The general notation for a standard form equation is Ax + By = C, where A, B, and C are coefficients, and the x and y are the same variables we’ve been looking at but in a different position from what we recognize.

To express in standard form, you simply just rearrange the y = mx + b form such that you have x and y on the same side, equal to a number. Let’s look at some examples:

Given that y = 3 x+ 5, standard form of this is 3x – y = (-5).

Given y = (1/2)x -15, standard form of this is (1/2)x – y = 15… also, if you don’t want to have any fractions in your answer, you can multiply everything by the number in the denominator, such that we now get x – 2y = 30. Both expressions mean the same thing and will produce the same line. (In fact, convince yourself that no matter what you do to the equation, so long as you do it to both sides, the line is the same. eg. Multiply it all by 100, you get 100x-200y=30000… looks different, but it’s not! Reduce it down and see for yourself!)

For graphing standard form equations, you still might want to go from standard form to the mx+b form, for which you may need to do a bit more math, but it’s still quite straight forward.

Given 5x – 15y = 10, you just have to rearrange things to get y by itself on one side:

(-15y) = (-5x) + 10

y = (1/3)x – (2/3)…

and then you can see it is a line with slope 1/3 and y-intercept (-2/3).

Both types of equations mean the same thing. They are just expressed differently, and y=mx+b gives immediate information about the line without having to do a lot of work. However, you should be able to use both forms interchangeably. Convince yourself that graphing standard form equations will give you the same line as graphing y=mx+b equations. They just look different because the numbers are rearranged. This should be obvious because if you start with a standard form equation, and convert it to y=mx+b and graph it, you have only rearranged things not added or removed anything. You do not have a new line.

Also, from these equations, you should be able to tell that whenever you have an equation with 2 variables (x and y), and there aren’t any exponents on either term, then you are dealing with a straight line. So while an equation in standard form may not immediately look like a straight line equation to you until it looks like y = mx + b, because it has an x and a y in it (without an exponent… exponents make the graph do cool things later), it is automatically a straight line!

The domainis simply all the values of x that a function can take, whereas the rangeis all of the values of y that the expression can have. A different way to think of this is to consider that domain is horizontally where the graph is located. If you were to take a graph and squash it down to a single horizontal line, what values on the x-axis would it include? Similarly, the range is vertically where the graph is located, and if you were to squish a graph from the left and the right into a single vertical line, what values on the y-axis would it include?

Another important problem you will have to work with is to determine if a given graph is a function or not. This is also a much simpler question that many people first think. The key to solving this type of question is to use the vertical line test (take a look at Khan Academy for a good explanation of this test). If you have a function and you substitute in a value for x, you will only get one y value. A function won’t give you more than one value for y. If it does, it isn’t a function at all! The vertical line test is a very quick and visual way of working with this principle. To use it, you simply draw a vertical line through any point of your graph. If the vertical line only passes through the line at a single point, then it is a function. That is, the value of x corresponds to only a single value of y. However, if the vertical line passes through more than one point (for example, a circle), then you do not have a function because for a value of x, there is more than one value of y.

So, now you know how to use the vertical line test to determine if your graph is a function or not. Once you have decided that your graph is actually a function, you can then move on to finding the domain and range of a line. Assuming that your line is plotted on a graph paper already with labeled points, finding the domain of a graph is incredibly easy. All you have to do is identify the horizontal ends of the line, and say that the domain is between the left point and the right point. The same strategy can be used to find the range of line graph. An important note to make is that sometimes a domain or a range do not have end points, and so we say that they extend to infinity.

Domain and range of f(x)=2^x

For example, if I asked you to find the domain of a line f(x) = x + 1, you can easily graph a straight line with a slope of 1, a y-intercept of 1, and the line extends forever up to the right, and down to the left. This particular line has a domain of negative infinity to positive infinity. So does the range. However, if I just drew a straight horizontal line, starting at the y-axis and extending to the right to x = 5, then you would say that the domain is between 0 and 5.

Another point to make is about the notation used to describe the domain and range. If the domain includes all of the numbers up to a point INCLUDING the point, then you use a square bracket [ or ] to represent that. On the other hand, if the domain includes all values approaching a point, but NOT the actual point itself (you will find this situation more when working with limits and calculus), then you use round brackets ( or ). So, for the previous question, I would record the domain as [0, 5] to indicate that the line extends from 0 to 5 and includes both of those points. If it didn’t include one, or either, of the points, I would use a round bracket next to the particular number. On the graph, if the line only approached a point, this would be represented by an open point, like an ‘o’.

When you advance to higher levels of graphing in algebra or calculus, finding the domain of a function will be a little more complicated, especially when you are looking at points that are irregular and cannot be precisely identified simply by looking at them. Piecewise functions are also a concept that will really require you to understand domains. However, I hope that by having domain and range explained here, it has given you at least a basic understanding of these very important math concepts, and with practice you will become an expert at finding the domain and range of a function.

Stretching and compressing graphs vertically is determined by the coefficient in front of the x (or more specifically, in front of the other direct modifications to x). Let’s look at a basic example: f(x) = x², a standard parabola.

Now, to vertically compress this curve, you put a ‘fraction coefficient’ in front of the x component of the graph. i.e. f(x) = (1/2)*x². This squashes the graph down by a factor of 2. Or, another way to look at it, every y value in this curve is 1/2 of the value in the starting curve. Plot your own points to convince yourself of this. Note that the curve crosses (-2,4) and (2,4) in the original curve, whereas the new one crosses at (-2,2) and (2,2).

Now, as you may guess, if you put a whole number coefficient in front of the x term, you will be stretching graphs. For example: f(x) = 2x²looks like this:

You can see that this has caused the parabola to stretch upwards. Note that it now crosses (1,2), not (1,1). Or once again, to look at it from a different angle, every y value is now twice the value as in the original graph. Compared to the original version, this is a vertical stretch graph.

The only other thing that you should keep in mind is that the coefficient to stretch or compress the graph vertically MUST be in front of any brackets that might be surrounding x, and the coefficient will act on any horizontal translation component and the exponent. Convince yourself of this by looking at graphs such as:

f(x) = (x-3)²……. and f(x)=2(x-3)²

f(x) = (x+1)³…… and f(x)=1/2(x+1)³

f(x) = x + 5………..and f(x) = 4x + 5

f(x) = (x-1)⁵ + 7…… and f(x) = 4(x-1)⁵ + 7

So far, all that I have talked about explains the concept of vertical compression and stretching. But, let’s now consider how we would go about showing a horizontal compression or stretch. If we want to squash a graph together along the x-axis, what would we do? Well, for any value of x, we would want the graph to be at some fractional value of that x. So mathematically, this means that we have to modify the x value. To perform a horizontal compression or stretch on a graph, instead of solving your equation for f(x), you solve it for f(c*x) for stretching or f(x/c) for compressing, where c is the stretch factor. The simplest way to consider this is that for every x you want to put into your equation, you must modify x before actually doing the substitution. Let me show you.

Consider again the parabolaf(x) = 2x². If we want to start plotting this graph, we could start by building a table of values and solving for f(1), f(2), f(3), etc. Doing the quick math, you can see that f(1) = 2. Now let’s say we want to horizontally compress this graph by a factor of 10. In this case, we do the modification on x before subbing in, so we apply the f(x/c) for compressing and see that f(1) becomes f(1/10). You then substitute 1/10 in for x, and solve. You then come up with the ordered pair of (1, 1/50). This is a ten times compression along the x-axis. If you draw some of these out, you will easily see that squishing or stretching along x is different from squishing or stretching along y.

Continuing with this same example, say we want to graph a horizontal stretch of a factor of 4. We then solve f(c*x), giving us f(4*x). For x = 1, f(4*1) becomes f(4), which works out to y = 32. Again, it is easy to see how much we have stretched the graph by this simple modification.

As you can see, stretching graphs (or compressing them), both vertically and horizontally, really isn’t that difficult. There are obviously a few things that you need to remember, especially the distinction between horizontal and vertical changes. But these modifications are just an extension of what you already know, building on your knowledge of horizontal or vertical shifts. Keep practicing, and you’ll get it in no time.

Shifting Graphs – Horizontal Translation

Now, to shift a graph horizontally, you include the shift amount with x. So, whatever action was being done just to x before, now you do that same thing to x plus the shift amount. Make sense? Probably not.

Check out the example below that hopefully demonstrates this better than I can explain with words.

If you want to shift the original function of f(x) = x² + 4 by 3 units, it becomes f(x) = (x-3)² + 4.

Can you see what I mean by including the shift amount WITH x. The ‘square’ function acts on the entire (x-3) term. This will cause the graph to shift 3 units to the RIGHT. This may seem somewhat counter-intuitive, but it is correct. Subtracting terms from x shift the graph to the right, whereas adding terms to x will translate them to the left.

In this example, x-3 causes a horizontal translation of the graph 3 units right… if it were x+3, it would translate the graph 3 units left. Here is a bit of a trick you can use to help you recall the direction of the shift caused by the signs. It may be easier to remember this by analyzing the “x and shift amount”, letting this small term equal to 0, and then solving for x. The result will show you how many units to move, and in what direction. Like this:

x – 3 = 0

x = 3 (shift 3 units right)

OR

x + 3 = 0

x = (-3) (shift 3 units left)

That shows you how far over, and in what direction, the new x values are! Technically, this is a way of finding a zero of the graph, but that is another post for another day. For now, I think it’s a helpful trick to apply at this stage!

I hope these postings on graph manipulations are helpful. Horizontal translation of functions and their graphs is still quite simple, albeit with the trick with the signs that you don’t have to worry about with vertical translation.

Let’s look at a simple equation first:

y = x

This equation represents a line that goes through (0,0), (1,1), (2,2), etc. Now, what if I asked you to draw the graph represented by the new equation y = x + 2. Some students may suggest to make a table of values and then plot the points on the graph… never a bad way to do things, but not necessarily the easiest.

Let’s think about things SLIGHTLY differently. In the first equation, let’s think about y specifically as being the height (and not just a different variable), and so the height for a given x value will always be equal to that x value. Not a huge change in our thinking, but it may help some to see graphical changes easier. Now, let’s apply this thinking to the second equation, and we see that the height is always going to be 2 greater than the x value.

Now, let’s spice things up a bit. Let’s look at a function. (The solution to the function for each value of x is similar to what y would be for each value of x in a regular equation).

f(x) = x²

You may recognize this equations as being a basic equation that describes a parabola that opens up. The lowest point on the graph is (0,0).

Now, if I said to draw the graph of f(x) = x² + 1, apply the same type of logic, but keep in mind one VERY IMPORTANT THING. The “change in height” is dictated by the single number that is not associated with the x variable… this may be clearer after another example, but let us focus on this one for the moment. You should hopefully be able to see that this change in our graph will result in a shift up of 1 unit, shown by the red curve below alongside the original blue one from above.

One final example, hopefully to clear things up for good. Consider a more complicated function, alongside a slightly modified version of it:

f(x) = (x-4)³ and f(x) = (x-4)³ + 5

So, after those examples, here is the takeaway message that hopefully is the ray of light for you.

Picture all the complex stuff that is happening to x as being one “chunk” of the height component, and then when you add the “+ 5” to the equation, you are really just adding an additional “height chunk” to the total height for a given x. So in the end, the second equation above looks exactly like this one, only shifted 5 units up:

Remember to combine terms if necessary, so that you are left with a single number to add to the x term (and whatever operator is acting on it). You will quickly find that vertical translations of graphs are far simpler than they may sound at first!

Why is “m” a symbol for slope?

So, why does m represent slope in math? Some records indicate that it first appeared in print in the mid-1800’s, in a geometry paper by a British mathematician named Matthew O’Brien, though other records suggest it dates back even further to Italy in the mid-1700’s by Sandro Caparrini. There is also a suggestion that it originated in the US. One theory suggests that because “slope” used to be called “the modulus of slope,” they shortened this to just “m”, though it is difficult to find evidence to back this up. Similarly, the m may come from the words for “mountain”… in Latin it is mons and in French it is montagne. There also is no evidence to support the myth that the French word for “to climb,” monter, provides the “m.” To dispute these suggestions, however, is the notion that the famous French philosopher Rene Descartes never recorded slope as m. One would think that if the origin was from a French word, a noteworthy French scholar would have used it! An alternate theory, which seems rational to me, is that the mathematician M. Risi suggested that the early letters a, b, c, etc are used to represent constants, the later letters z, y, x, etc are used to represent unknowns, and the middle letters represent parameters. Therefore, since slope is considered a parameter, it may have arbitrarily been assigned this mid-alphabet designation. For all we know, it was a practical joke that got taken too far and is now an accepted concept of the mathematics of graphing!

Whatever the true origin of this seemingly strange symbol, it won’t provide any insight into easier ways to work with slopes, lines, and graphing! So, the next time someone asks this question, you don’t have to feel bad to say “I don’t know and I don’t think anyone else does either. Now go do your homework.

But first, let’s recap!

In general, a piecewise function may be considered as a function that is described by more than one equation, with each equation only being applicable to the function over specific domains. So, this is like saying you have a line that rises to the right on a graph across the domain of x = 0 and x = 3, and then the line falls to the right from x = 3 to x = 6. The upward sloping line is described by one equation, and the downward sloping line is described by its own different equation. (A point of interest in this example is at x = 3, where both equations apply!)

Together, these two lines form the graph of our function. (I realize that the vertical axis should be labeled as f(x) rather than y, though I can’t seem to find the setting to change this in my graphing program…) A seemingly complicated mathematical curve like this is bound to be expressed as a complicated looking set of equations, right? Well, as I outlined in my introductory post, it’s not that bad when you know what you are looking at.

You basically express a piecewise function as a list of equations, grouped one on top of the other to the right of an open parenthesis.

At first it looks complicated, but it is really just two equations, with their domains explicitly noted. (Actually, piecewise functions can have many equations in a list, though I will just work with two for simplicity!) In this example, you can see that there is a positively sloped line that spans the domain of 0 to 3, and a negatively sloped line that spans the domain of 3 to 6. In this case, as I have created this question, the vertex of the graph is a solution to both equations, which I indicated in the problem as the greater than or equal signs. So, if you want to know what f(3) is, you need to assess the piecewise components to see which equation you need to solve in order to arrive at your solution. In this case, both equations are valid when x=3, so you can solve f(3) using either one. You will find that indeed each equation will provide you with the same output.

That was a fairly standard example of graphing a piecewise function. However, you can also have a piecewise function that isn’t continuous.

In this case, you can see that my function is composed of two line segments. The first part is the line f(x)=3, over the domain from zero to 3. However, pay attention to the greater than signs! Here, the lower limit is greater than or equal to 0, though the upper limit is where x is less than (not “or equal to!”) 1. In situations like this, you denote this on the graph with an open circle around that point. This implies that you are including all of the points along the curve up to the precise point that is specified. So, in this graph, when x =1, you have to abide by the second equation, because that is explicitly given in the domain of the second equation. However, at x = 0.99999999999999, you fall within the domain of the first equation. So, pay attention to your signs! If your test question here said “what is f(1)?”, if you answer as f(1)=3, you would be wrong! It is a simple mistake to make, so it is best to really pay close attention! Making the mark on the graph will help you to not get mixed up by simply looking at the signs in the expressions.

Another graph notation to keep in mind is to explicitly express points on the line as “greater than or equal to”. You can do this by marking with a filled circle. Often, you won’t actually have this shown, as it frequently is just understood that if there is no open circle, the point is included (as I have done above). However, it is wise to understand that there are two ways of marking points on a graph – a filled circle includes the point, but an open circle means getting infinitely close to the point but never exactly including the point. These are important notes and are favourites for teachers to put on exams!

A final brief note I will include here is a refresher on the convention for expressing domains (also applicable to ranges), known as interval notation. As I show above, the first part of the expression is f(x) = 3 when x is greater than or equal to 0 and less than 1. This is awfully wordy for mathematicians! So obviously, there is a better way to denote this, and it is done by using brackets. The convention is that if the greater than (or less than) includes “or equal to”, you use square brackets, like this: [ ]. If there is no “or equal to” then you would use round brackets, like this: ( ). The left bracket refers to the lower/left limit of the domain, whereas the right bracket refers to the upper/right limit of the domain. Inside the brackets, you include the lower and upper limit of the domain, separated by a comma. So, for this example, we could simply express the domain of this first part of this piecewise function as [0, 1). Similarly, the second part of the function can be denoted as [1, 5]. Of course, both brackets do not have to be the same – each one is completely dependent on what the sign is telling you. So you can have [ ], ( ), [ ), or ( ]. This is a great shortcut to help you to express your domains and ranges, and it really comes in handy when graphing piecewise functions. The only piece of advice I can add is make sure your square brackets don’t look like round ones and you get confused!

When I introduced functions, I referred to the standard notation as f(x). However, you will quickly come across problems that use different letters – this means the same thing. You will probably see things like g(x) or h(x), and they are just used to differentiate between different functions. For example, consider the following functions:

f(x) = 2x + 3

g(x) = x – 8

h(x) = 4x + 1

This shows that the functions f(x), g(x), and h(x) are equal to the given expressions. So f(x), g(x), and h(x) all represent different functions (continuing the metaphor from earlier, these are all different “machines” that act on your input to produce their specific outputs). The f, g, and h are all essentially saying the same thing – that is, they are the names of their function. It’s comparable to saying 3x = 12, or 3a = 12… the underlying math is the same, though in this case, the name of the variable is different. Same thing with function names.

So, now that you know how to differentiate between and name different functions, we can do slightly more complicated things – things like complex functions. If for the following function we want to solve f(1), we substitute 1 in for x, and solve it to get 5:

f(x) = 2x + 3

f(1) = 2(1) + 3

f(1) = 5

However, we can now ask for something more complex such as f(g(x))… read as “f of ‘g of x’.” Looks complicated, but keeping in mind what the function notation is saying and what a function is actually doing, we can easily solve this by substituting the expression for g(x) in for x (instead of just subbing in a single number like before). Continuing our metaphor, it’s like saying that our function machine f(x) has an input of the entire machine g(x).

f(x) = 2x + 3, g(x) = x – 8

f(g(x)) = 2(x – 8) + 3

f(g(x)) = 2x – 16 + 3

f(g(x)) = 2x – 13

Furthermore, if we wanted to solve f(g(5)), we can do one of two things. We can evaluate the expression as we just did, then solve when x= 5 (like I said above, this is like giving the f(x) machine the entire g(x) machine as its input. It’s a machine working on a machine working on a value!). Or, we can solve g(5) first, then substitute that into f(g(x)) – basically, starting at the inside and working your way out to the outermost function (that is, we provide the g(x) machine the input of 5, get the output for that, and then give that to the f(x) machine as its input). Both methods will get you the same answer, and it’s really up to you to do whichever you are most comfortable with (unless you are instructed to specifically do one way or the other!).

Method 1:

f(g(x)) = 2x – 13

f(g(5)) = 2(5) – 13

f(g(5)) = (-3)

Method 2:

g(x) = x – 8

g(5) = 5 – 8

g(5) = (-3)…

then…

f(x) = 2x + 3

f(g(5)) = f(-3) = 2(-3) + 3

f(g(5)) = (-3)

Both methods get the same answer, because they are doing the same things. It’s just a matter of what you are more comfortable with. Function notation can get very confusing like this, but for problems like this, you just have to basically work from the inside out, just as in any other math expression you’ve seen! (eg. (2x + 7(3x -4) + 2)…. solve the inside bracket first, then the outside). They may seem tricky, but you will be surprised at how easy they are when you see through the notation and know what they are asking!

Now that I’ve explained complex functions, I’d like to expand on this to discuss what is known as piecewise functions. I’ll post a follow-up shortly to focus on graphing piecewise functions, but here is an introduction to them.

The functions that I’ve discussed so far (and generally, what most people typically talk about) can be drawn as continuous, smooth curves with no gaps. These tend to be described by a single equation. But what if you consider a graph that is composed of two distinct sections – such as one with a rising diagonal line attached to a horizontal line at the top? Taking each section on its own, you can imagine how it would have its own unique equation. However, combining these sections into graphs such as these, which are described by multiple equations each applying to specific domains of the entire curve, these are called piecewise functions. Obviously, they get their name from the fact that the function is made up of several “pieces”. Here is an example of what a piecewise function would look like on a graph, as well as how you would express this type of function mathematically:

The notation basically says that here is the function f(x), and it is equal to the listed equations over the specified domains. So here, f(x) is equal to 2x-2 only across the domain that is between 1 and 2.5. Similarly, f(x) is equal to 3 when the domain is between 2.5 and 6. In this case, at the point on the curve corresponding to x = 2.5, both equations are valid and will produce the same value. However, it is important to pay attention to the “greater than” and the “greater than or equal to” signs, depending on what your particular question is talking about.

With that, I will conclude my brief introduction to complex functions and piecewise functions, and I will look ahead to my next post where I will go into a bit more detail about graphing piecewise functions and some things to keep in mind, as I have hinted at above.

First, you need to recall what exactly is meant by “function.” In my introductory post, I showed you how I was first taught this concept, by using the metaphor of a machine. You first start off with your input number, which you feed into the machine (which means, you perform the function calculation on it), and then the machine provides you the output when it has done its work. An important point about this metaphor is that every time you put in a specific starting number, the machine will always give you the same output number. An equally important point is that for every number that you give your machine, not only does it give you the same answer each time, it gives you only a single output number. So, for example, if you feed the function an input value of 2, every time you do the calculation, you will always get a 10 and only a 10. This is a property of functions. If your calculation gives you more than one output value, then there are two options – either you made a mistake, or you are not dealing with a function at all! This is where the vertical line test comes in handy.

So then, let’s begin to relate this concept to a graph. When I talk about providing a number for our machine, I am referring to providing a value for x to our function f(x). The machine/function then produces an output, corresponding to the value of f(x) at x – though, for this comparison, it may be easier to just think of the output as a y-value. So, if we input an x-value, and get out a y-value to correspond with that x, we have determined an ordered pair coordinate! And as I stated above, our machine/function will produce only a single result for every value of x that you provide it.

The vertical line test definition is a check to see that your graph only has a single y-value for every x-value in its domain. To do this, all you need to do is scan over all of the x-values on your graph and make sure that there are no more than one y-value for each. To easily accomplish this, all you need to do is take your pencil, line it up vertically against your graph, and then move it from side to side, making sure that the pencil is never touching the line at more than one point at a time. The vertical line test is just that easy! If your pencil only ever crosses your curve at one point at a time, you have a function. As a very common example of what is NOT a function, consider a circle and notice how it fails this test.

And that’s all there is to know about the vertical line test. Now you are armed with the knowledge to be able to simply look at a graph and decide whether it is a function or not! For a couple of vertical line test worksheets, check out this page over at Math Warehouse.

So then, let’s start at the beginning. What does it mean when your teacher asks you to find domain? The domain of a function is simply all of the values of x that a function can have. (I guess you can kind of think of this in terms of a map after all… the domain includes all of the x values where your curve lives.) A very important point to understand with this definition is that it strictly refers to the values of x for your function, no matter the corresponding y values. Consider it as where horizontally your graph sits on the axes. With that in mind, here is a neat way of visualizing this. Since the y values don’t matter, pretend that you can take your graph and squash it down so that it is nothing but a horizontal line. Now if you look at this squished version of your curve, you can simply tell what the domain is by where this line sits along the x-axis. This is a very basic way of looking at this concept, but hopefully it does a good job of introducing it.

Moving along to the other half of our topic, now that you know the definition of domain, range must surely be something similar, right? If we were just talking about x values, maybe range deals with y values? Of course it does! The range of a function is all of the values of y that a function can have. Similarly, you can consider this to be where vertically your graph sits on the number lines. Again, you can try to visualize this better by squishing your graph from the left and right into a single vertical line, and then you can tell what y values should be included.

So with those definitions, you should hopefully be able to solve some of your math questions. However, I am going to mention a few other things that you need to consider when working with problems such as these.

The first point you need to assess is whether a graph that you are given is actually a function or not. Not every graph is! Naturally, you may think that now there is going to be some kind of weird calculation or something you have to do to determine this. Well, there is, but it’s not nearly as complicated as you think! To decide if a graph is truly a function, it simply has to pass the vertical line test. And yes, that is as easy as it sounds! Take a look at your graph, and place your pencil vertically on it (or draw a bunch of vertical lines through your graph), in the direction of the y-axis. As you move your pencil left and right, does it only ever cross the line at one point at a time, or does it cross more than once? If more than one point on the graph ever touches your vertical pencil at the same time, it is NOT a function. If your vertical lines only pass through the curve at a single point, then it is a function.

What does the vertical line test actually tell you to allow you to have confidence that you are truly working with a function? It demonstrates that for any value of x that you input into the expression, you will only ever get a single value for y. A function always has one y value for each x value. Imagine a straight line: for every x there is a corresponding y. Now, compare to a more advanced curve, such as a circle. A circle would obviously fail a vertical line test, because you can easily see that you get two y values in its range for each x value in its domain (except at the extreme ends). As simple an object as you may think a circle is, it actually is not a function for this very reason.

So that is the first part addressed: determining if your graph is a function in the first place. Having decided that it actually is one, you can then move on to finding the domain and range of a function. This step can actually be remarkably simple, or require a little bit of work. In the simplest case, you will have a continuous curve on a graph (no gaps!) with labeled points, and you can simply read off of the graph what the x values are on the far left and right side of you line, and then state that domain contains all x values between those two points. Equally, you can do the same sort of thing with the y values. Look at the highest and lowest points on the graph, and assuming no holes in the curve, your range is going to be all values contained between those two extreme points.

I should add a very important note here. Not all functions have end points. In fact, unless it is specifically defined in the expression, you may say that most functions do not have end points. Instead, we say that they extend to infinity. As such, it is not uncommon to say that a domain or range is from negative infinity to positive infinity. Consider a line such as f(x) = 3x. You could substitute values into x such as -13848, 4892/38577, or 10000000000, and then calculate the corresponding y. The line extends both left and right, and up and down, forever.

f(x) = 3x

On the other hand, some functions may be defined to purposefully limit their ends. This would be the case when given the expression f(x) = x + 2, for x < 0. In this case, the equation explicitly states that the curve only exists over the positive values for x. So, we would say that its domain ranges from 0 to positive infinity. For the range, in this case, your lowest point is your restricted x value, so you can plug that into the equation to give f(0) = 2. Therefore, your range includes all values from 2 to positive infinity.

f(x) = 2x, x > 0

Now, what about if the equation of the curve is more complicated. What do we do if we have something like f(x) = (2x + 1) / x. It doesn’t look that hard. There isn’t any rearranging or algebra that needs to be done. You can produce a table of values very simply by subbing in many values for x and determining the y values, and then you can plot the curve. When you do that, you will see that the graph looks more complex that the continuous lines we’ve seen so far.

f(x) = (2x + 1) / x

You can see that as the curve approaches x = 0, there doesn’t appear to actually be point precisely at x = 0. Instead, the curve appears to be asymptotic to the line x = 0, and similarly at the line y = 2. This is a result of the curve being undefined there. This may sound complicated, but it’s not at all. A curve is undefined when the denominator of your equation is equal to zero. This is because you cannot divide something by zero. Try it on your calculator. Punch in any number, divide it by zero, and it will return an error. In this case, you can see from the graph that the domain includes all numbers except 0, and that for the range, it appears to include all numbers except y = 2. If you were to only look at the equation, you could also equally say that the domain includes all numbers except zero, because only by putting zero in the denominator would your equation be undefined. To determine the range, you would need to do some rearranging to express it in terms of y. In this case, you would come up with the rearranged formula x = 1 / (y – 2). I’ll leave this for you to work through and verify. Going back to our rule that you can’t have a zero in the denominator, this curve is undefined when you just say that y – 2 = 0, leading to y = 2. And comparing this to our curve, you can indeed see that this is the case.

I should also discuss graphing conventions and proper notation for expressing domain and range, because there is a connection between the two. If you have a short line segment, say f(x) = 2, extending between and including the points at f(1) and f(5), this curve obviously does not extend to infinity.

f(x) = 2

Because its end point include those points (i.e., 1 < x < 5 ), you would note this on the graph with a solid dot. (Unfortunately, I haven’t figured out this functionality on my graphing program, so take my word for it!) However, if I said that the domain extended between those two points but did not actually include them (i.e. 1 < x < 5), you would represent this with an open dot (like an ‘o’) at those points. Pay attention to what is happening and this points, because this now affects how you express the domain. You denote domain and range by using either round or square brackets, depending on if you include the point or go right up to the point but don’t include it. So, if you have a domain of 1 < x < 5, you can write this as [1, 5]. Similarly, if your domain is 1 < x < 5, you could write (1, 5). Don’t get these confused as being ordered pairs! You can also mix and match these brackets, so you can have something like (1, 5] as well. Importantly, if you are stating that your domain or range includes infinity, infinity always gets round brackets. This is because you cannot include the “infinity point” precisely. You are always approaching it.

If you are given a piecewise function, determining the domain and range of that can be a bit trickier. You need to pay attention to all parts of the expression that you are provided, and make a note of anywhere that any of them are undefined. Treat each part individually. Also note the domains and ranges over which each part is defined, and whether there are any gaps in the line.

As you get to higher levels of graphing in algebra and calculus, things will get more complicated. You will have curves that are irregular that require you to analyze their equations. Determining precise points on them will require analysis of their equations, and you won’t be able to identify them merely by looking at the curves. You will undoubtedly find problems where you wish you had a domain and range calculator, but all of these problems are solvable if you approach them the right way.

I hope that this post has explained this very common and rudimentary graphing concept in a way that is understandable. It doesn’t need to be a complicated as some students think, nor as some teachers make it! Practice with easy examples until you are comfortable enough to move on, and before you know it, you will be an expert at finding the domain and range of a function, and you won’t need help with math anymore.