The inverse of a function is found by reversing the order of the coordinates in each ordered pair.
So for instance, suppose I had the function f that's defined by these three ordered pairs right here: (1,5) (2,7) (3,9).
We find the inverse of f by reversing the order of the coordinates in each ordered pair.
So (1,5), when it's reversed becomes (5,1); (2,7) becomes (7,2); and (3,9) becomes (9,3).
Now, this set of ordered pairs right here, which is different than this set right here, is the inverse of my original set.
So, if f is this set of ordered pairs, the inverse of function f is this set of ordered pairs, right here.
Now, we we're not usually given a set of ordered pairs to define a function; usually, we're given a rule that tells us how to get the ordered pairs.
Now, if f is given by, let's say the equation y = 3x - 1, we can find the equation of the inverse
by simply interchanging the variables x and y in the equation; and then, we'll have the equation of the inverse of f.
So, x = 3y - 1 will give me ordered pairs in which the coordinates have been reversed, from the ordered pairs in this original function,
right here. So, all functions have inverses, some of which are functions and some of which aren't, we'll get into that in a minute,
but the inverse of a function is the set of ordered pairs with the coordinates reversed. Now,
if we're given the equation of a function, we can find the equation of the inverse by reversing, by exchanging the variables x and y in the equation.
Let's look at an example. For example number 1, I have f(x) = 3x - 1, and I want to find the inverse.
Well, I'm going to go back to my notation for y, and that is, I'm going to write y = 3x - 1.
Now, to find the inverse, okay, the inverse is x = 3y - 1. So, this is my original equation, right here.
Now, this is the equation of the inverse; and I find the equation of the inverse by simply exchanging x and y in my original equation.
Next, let's solve this for y. So, I'll add 1 to both sides, and exchange the sides, so I get 3y = x + 1.
Add 1 to both sides; and now write the x + 1 over on this side, and the 3y on this side. Now, I'm going to divide both sides by 3,
so I end up with y = (x + 1)/3. So, this is the equation of the inverse of this function right here.
So, if I say y = 3x - 1, its inverse is y = (x + 1)/3. Notice that the function itself multiplies x by 3 and then subtracts 1.
The inverse adds 1 and then divides by 3, the inverse operations in the reverse order. Now, let's take a look at some ordered pairs;
and also I want to graph these two functions on a coordinate system, here. So, let's look at some ordered pairs.
First, I want to write down, I'll copy the functions down again for you; so, first of all here is my function, my original function:
y = 3x - 1. Now, let's find some ordered pairs: let's let x = 0; when x = 0, y = -1.
Let's let x = 1, and then I have 3(1) = 3, 3 - 1 = 2. And let's try one more; how about, a negative value of x?
What about negative, let's put in -2; 3(-2) = -6, -6 - 1 = -7.
Now, let's look at the inverse of f, I'm sorry the inverse function or the inverse relation, I'm not sure it's a function yet, although it is linear.
Now, let's substitute into this, let's substitute in x = -1. When x = -1, I have -1 + 1 = 0,
0 ÷ 3 = 0. Notice that that's the same as this ordered pair, with the coordinates reversed.
Let's see what corresponds to this ordered pair by substituting in x = 2 into my inverse: 2 + 1 = 3, 3 ÷ 3 = 1.
And down here, let me substitute in -7: -7 + 1 is -6; -6 ÷ 3 = -2.
So, you can see that, in fact, if I have my original equation right here, I exchange x and y and then solve for y,
I get the equation of the inverse of my original function right here. Now, this original function had these three ordered pairs among its solutions.
Now, I take the equation of my inverse, and I find three ordered pairs, and I find that they're the same ordered pairs that I had up here,
just with their coordinates reversed. So, in fact, reversing or exchanging x and y in the equation does reverse the coordinates
in all the ordered pairs, giving you the equation of the inverse. Let's look at the graphs of these two. First of all, let's graph this inverse right here:
y = (x + 1)/3. I'll use these two points: (-1, 0); and (2, 1). So, there's that point.
Now, let me draw a line between those two points, and see what we get here. Let's see, looks about like that.
Okay, so that red right there, that's the graph of the inverse. Now, let's graph the original function itself: (0, -1), that's right here,
and (1, 2). So, I go forward 1 and then up 2. So, x = 1, y = 2. Let's sketch the graph of that by drawing a line between those points.
Okay so, there's the graph of my original function right there: y = 3x - 1; this is the graph of y = (x + 1)/3;
and what you will notice is that if you were to draw in the line y = x, which would look just like that,
that my two graphs are symmetrical about this line y = x. So, if I draw in the line y = x you see that the two graphs,
if I reflect one of them about the line y = x, I'll get the other one. So, this graph and this graph have symmetry about the line y = x.
Okay so, that's one of the relationships that we have between the graph of a function and the graph of its inverse.
Next, what I want to do is graph a function, and then look at the graph of its inverse and see if I can tell by looking at that,
whether the inverse is itself a function. Here's our next example. Okay, for problem 2, I want to graph y = x² - 3 and its inverse.
Well, y = x² - 3 is a parabola; when x = 0, y = -3, so it goes through this point.
And when x = 1, I get (1)² - 3, that's -2; same thing will happen when x = -1.
When x = 2, I get (2)², which is 4 - 3, which is 1; same thing will happen when x = -2,
I get this point right here. So, I have these points: (0, -3); (1, -2)
(-1, -2); (2, 1); and (-2, 1).
Now, if I was to put in 3, I'd get 9; 9 - 3 = 6; so, that's this point right here; and now, we get this point also.
Okay so, here's the graph of y = x² - 3. Now, to graph y equals its inverse,
its inverse is going to be x = y² - 3. Now, I can graph that just by itself if I want, but I don't have to.
What I can do is take the line y = x, and reflect my first graph about this line; and I'll have the graph of the inverse, okay.
So, let's do that. Here's the line y = x. So, I draw in that line y = x, and now I'm going to reflect my blue graph about this line.
So, where I had x = 0, y = -3, I'll have (0, -3).
Here, I had x = 1, y = -2; what I'll get, when I reverse the order of those coordinates, I'll get
x = -2, y = 1; same thing down here. And then, I'll get this point right here, and also this point right here.
And then, if I go over to 3, I get 6; and in the -3, I'll get 6. So, here is the graph of the inverse of my original function.
So, it looks like if I take this blue graph and just reflect it about the line y = x, that is, if I folded the paper over that way,
this would be the graph that I would get. So, if I reflect the blue graph right here, about the line y = x, this point goes over here,
this point goes over here, this point comes here, this point right here goes there, this point goes over there, this point goes over here.
This is the graph of my inverse x = y² - 3. Now, by looking at the graph and knowing this vertical line test,
you can tell that the graph of the inverse is not itself a function. So, here is my original function right here.
The graph of its inverse is not a function. Now, why does that happen?
Well, that happens because my original function had some ordered pairs that, where I had the same second coordinate.
So, here, for instance, I had x = 1, y = -2; here's x = -1, y = -2.
It's still a function because I don't repeat first coordinates, but when I reverse the order of the coordinates in those ordered pairs,
I end up with these ordered pairs: (-3, 0); (-2, 1); (-2, -1); and see right away, I have two different ordered pairs
here with the same first coordinate, meaning that this graph right here is not going to be the graph of a function.
So, how can I be sure that my original function will produce an inverse that is also a function?
Only if my original function is what's called a one-to-one function, which means that it's a function, so it doesn't repeat first coordinates;
and it's one-to-one, which means that it doesn't repeat second coordinates. So, not repeating first coordinates
and not repeating second coordinates guarantees that your function will have an inverse that's also a function.
Now, for the next example I want to look at a function that's like that.
For problem 3, I want to graph y = 3ˣ; so, let's see, I've graphed this previously. When x = 0 y = 1; when x = 1, y = 3;
when x = 2, y = 9, that's way up here; when x = -1, y = 1/3; (2, 1/9); and so on, and so forth.
So, I get a graph that comes down and then crosses the x-axis, does not cross the x-axis but gets us as close to the x-axis as we want.
Now, I know that this function right here is a one-to-one function; it will not repeat any second coordinates, and I can tell,
because no horizontal line will cross this graph in more than one spot. So, the vertical line test you use to tell if a relation is a function or not.
If you can't find a vertical line that crosses in more than one spot, then you know that your function doesn't repeat any first coordinates.
The horizontal line test tells you that if you have a function, whether or not you have a one-to-one function or not;
so, if no horizontal line crosses in more than one spot, then you must have the graph of a one-to-one function.
Now, every one-to-one function has an inverse that's also a function; so, let's find the inverse of this graph right here.
First, I'm going to put in the line y = x, so let me do that first. And then, we'll have something to reflect about, or to reflect our graph around.
(Oops, moved my ruler.) Okay, so there's the line y = x. Now, I want to take my blue graph and reflect it about that line,
so, what I'm going to get is a graph that will go through this point, come up to here, and then go out like this; and it will come down,
and it will approach the y-axis like this. Now, can you see if I look down this line right here,
this graph and this graph are symmetrical about this line? This, then, is the graph of the inverse of y = 3ˣ, which is x = 3ᵞ.
Notice that, for my inverse right here, no vertical line crosses this graph in more than one spot, so this is the graph of a function.
I knew that would happen, though, because my original function was a one-to-one function, it doesn't repeat any second coordinates.
So, when I reflect it about the line y = x, I'm, going to get a function;
I'm going to get the graph of a function that passes that vertical line test because it doesn't repeat any first coordinates.
Now, this particular function and its inverse right here, which is an inverse function, are going to become important in the next chapter
when we look at logarithms. I want to do one more problem here, and that is to find the inverse of f(x) = (2x + 1)/(3x + 1).
So, I'm going to use my y-notation to write this as y = (2x + 1)/(3x + 1). Now the inverse:
the inverse will be x = (2y + 1)/(3y + 1). Now, what I need to do next is solve this
equation right here for y,
so I get y in terms of x; and then, I'll have the equation of the inverse. Let's do that. I'll multiply both sides by (3y + 1);
I get (3y + 1)x = 2y + 1. Now, let's multiply this out: 3xy + x = 2y + 1. Let's take all my y-terms,
and put them on one side: 3xy - 2y is equal to, and I'll add -x to both sides, -x + 1.
So, I've collected all my y-terms here, and all the terms without y over here. Now, let's factor a y out,
and I'll have left (3x - 2)y = -x + 1; divide both sides by 3x - 2,
and I have y = (-x + 1)/(3x - 2). Now, this is the equation of the inverse;
it turns out that my original function right here is a one-to-one function. This function right here, if we could sketch the graph of it real quick,
you would see that this is a one-to-one function. This function is not going to repeat any values of, any first coordinates or second coordinates.
That means that its inverse, which is given by this equation right here, is also a function. So when that happens,
when I have my original function is one-to-one, its inverse comes out to be a function, and I have inverse function notation that I use,
and I rewrite it like this: f with a little exponent of, which looks like an exponent of 1, it's not an exponent,
it's just notation, of f⁻¹(x) = (-x + 1)/(3x - 2).
So, this notation right here is what, is inverse function notation f⁻¹(x) stands for;
it's a notation that stands for the inverse of function f. I only use it when my original function is one-to-one;
then, I know that its inverse comes out to be a function itself, so I write f⁻¹(x) for that value of y. So, I have functions, f(x), and their inverse functions, f⁻¹(x), if the original function is a one-to-one function.