### Video Transcript

Graphs of Inverses of Functions

In this video, we will learn how to

use a graph to find the inverse of a function and analyze the graphs for the inverse

of a function. To begin, let’s consider what is

meant when we say the inverse of a function. A function 𝑓 can be inverted by

the function 𝑔 when 𝑓 of 𝑥 is equal to 𝑦 and 𝑔 of 𝑦 is equal to 𝑥. We can think of these two functions

as undoing the actions of each other. This means that if we were to apply

𝑓 and then we were to apply 𝑔, it would be the same as doing nothing and the

result would be our starting value. We can mathematically represent

this as a function composition as follows. Here we have considered a single

input value 𝑥 and a single output value 𝑦. However, if 𝑓 of 𝑥 is a

one-to-one function, it can be inverted over its entire domain.

We’ll return to this “one to one”

constraint later in this video. But for now, we should understand

that when 𝑓 of 𝑥 is a one-to-one function, we can find a uniquely defined inverse

function. Rather than calling this function

𝑔, it’s common to use the following notation. We would read this as 𝑓

inverse. We should also note that 𝑓 is the

inverse function of 𝑓 inverse. So the relationship goes both

ways. Returning to our function

composition, we can say that applying both 𝑓 and 𝑓 inverse in either order will

give us the same value that we started with.

Before moving forward, let’s

highlight one quick point about notation. We should be very careful not to

mistake this minus one for an exponent. As a very quick example, if we had

some function 𝑓 of 𝑥 equals two 𝑥 minus three, 𝑓 inverse would not be two 𝑥

minus three all raised to the power of minus one. This video will not go into any

detail on how to algebraically find the inverse of a function. But it’s enough to say this common

misconception should be avoided. Okay, now that we’ve recapped the

inverse of a function, let’s see how it relates to graphs. Consider some invertable function

𝑓. When graphing functions, we usually

consider the 𝑥-axis as the domain of the function, which we can think of as an

input. We consider the 𝑦-axis as the

range, which we can think of as an output.

Let’s say we were to input some

value 𝑎 into our function which gave an output of 𝑏. More formally, we might denote this

as 𝑓 of 𝑎 equals 𝑏. If we were to plot the graph of our

function 𝑓 of 𝑥, we would say that the point with coordinates 𝑎, 𝑏 lies on the

line or curve. Now let’s consider the function 𝑓

inverse. As we saw earlier, 𝑓 inverse

undoes the action of 𝑓. This means that if we were to input

the value of 𝑏 into 𝑓 inverse, the resulting output must be 𝑎. We can follow a similar line of

reasoning to conclude that the point with coordinates 𝑏, 𝑎 must lie on the graph

of 𝑓 inverse.

Let’s consider what we have just

found. If we find a coordinate 𝑎, 𝑏

which lies on the graph of 𝑓 of 𝑥, we can reverse the ordered pair to find a

corresponding coordinate 𝑏, 𝑎 which lies on the graph of 𝑓 inverse. We can also think of this as a swap

in the 𝑥- and 𝑦-coordinate. Since we followed a general method

to get here for any given point on the graph of 𝑓 or indeed of 𝑓 inverse, this

relationship must be true.

Another interesting point, since

this relationship is true for all values, we might be able to see that the domain of

𝑓 is the same as the range as 𝑓 inverse and the range of 𝑓 is the same as the

domain of 𝑓 inverse. Let’s now pick some values and see

what all of this looks like on a graph. Consider the function 𝑓 of 𝑥

equals two 𝑥 minus six. If we wanted to pick some values to

plot in the top-right quadrant for 𝑓, we could use the coordinates three, zero;

four, two; and five, four. Now remember, to find points on the

graph of 𝑓 inverse, we can simply swap our ordered pairs. This means that the points zero,

three; two, four; and four, five all lie on the graph of 𝑓 inverse.

At this point, we might begin to

notice a pattern or perhaps even a link to function transformations. The change that we have described

with the ordered pairs switches the 𝑥-coordinate for the 𝑦-coordinate and vice

versa. This change actually corresponds to

a reflection in the line 𝑦 equals 𝑥. Looking at the graph, we can see

that all three of the points we’ve plotted display this rule. Another thing we might notice is

that if our function 𝑓 intersects the line of reflection 𝑦 equals 𝑥 at any point,

𝑓 inverse will also pass through this point. On our graph, this happens at the

point six, six. Joining up all of our points, we

should be able to see our reflection more clearly.

Again, note that this video will

focus on graphs of inverse functions rather than algebraic manipulation. The important rule that we have

just found relating to this is that inverse functions have graphs which are

reflections of each other in the line 𝑦 equals 𝑥 and the coordinates of the points

on these lines or curves are reversed ordered pairs. Let us now take a look at an

example of a question.

The following is the graph of 𝑓 of

𝑥 equals two 𝑥 minus one. Which is the graph of the inverse

function 𝑓 inverse of 𝑥?

And just to be clear for this

question, 𝑓 of 𝑥 is the blue line, and our options for 𝑓 inverse of 𝑥 are shown

here. When trying to find graphs of

inverse functions, we should remember the following important rule. Inverse functions have graphs which

are reflections in the line 𝑦 equals 𝑥 and have corresponding coordinates which

are reversed ordered pairs. One approach to solving this

problem would be to draw the line 𝑦 equals 𝑥 onto our diagram. We would then reflect the graph of

𝑓 of 𝑥 using the line 𝑦 equals 𝑥 as our line of symmetry. Note here that the question has

tried to throw in some confusing factors. For the graph given by the

question, the 𝑥-axis goes from negative four to four and the 𝑦-axis goes from a

negative six to six. This is not the same for the axes

in all of our options.

In spite of this, we can still

begin to eliminate some options. We can first observe that the graph

of 𝑓 inverse of 𝑥 has a positive gradient. Looking at options (b) and (c), we

can see both of these have a negative gradient. They can therefore be eliminated as

they cannot be the graph of 𝑓 inverse. Next, we can observe that the graph

of 𝑓 inverse appears to have a positive 𝑦-intercept. Of our remaining two options,

option (a) does have a positive 𝑦-intercept, whereas option (d) does not. This means that we can eliminate

option (d). This means that the answer to our

question must be option (a).

We might also want to consider that

there are other ways that we could have solved this problem. In particular, we can use the fact

that corresponding coordinates are reversed ordered pairs. Imagine we had not reflected 𝑓 of

𝑥 in the line 𝑦 equals 𝑥. Instead, we could eliminate the

incorrect options by cleverly picking a point on the graph of 𝑓 of 𝑥. Let’s pick the 𝑦-intercept which

is the point negative one, zero. Reversing this ordered pair gives

us the coordinates of a corresponding point which we know must lie on the line of 𝑓

inverse of 𝑥. That coordinate is zero, negative

one. We can then use this information by

looking at all of the options for the graph of 𝑓 inverse of 𝑥 and seeing which of

these graphs appears to pass through the point zero, negative one.

Of course, we already know the

answer. The graph of option (a) does pass

through the point zero, negative one. The graphs of option (b), (c), and

(d) do not. Again, we have confirmed that the

graph of 𝑓 inverse of 𝑥 is option (a).

Let’s now take a look at another

example.

Shown is the graph of 𝑓 of 𝑥

equals five 𝑥 cubed plus six. Find the intersection of the

inverse function 𝑓 inverse of 𝑥 with the 𝑥-axis.

To begin, we might remember the

following rule. Inverse functions have graphs which

are reflections of each other in the line 𝑦 equals 𝑥. One approach to solving this

problem might be to draw the line 𝑦 equals 𝑥. We could then reflect the graph of

𝑓 of 𝑥 using 𝑦 equals 𝑥 as our line of symmetry and then find the intersection

of 𝑓 inverse of 𝑥 with the 𝑥-axis. This alternate approach relies on

the fact that coordinates of points which lie on the graphs of inverse functions are

reversed ordered pairs. In other words, if we swap the 𝑥-

and 𝑦-coordinates for a point on the graph of 𝑓, we get the coordinates of a

corresponding point which we know is on the graph of 𝑓 inverse. Also note that this relationship

works both ways.

Okay, so we’re looking for the

intersection of the graph of 𝑓 inverse with the 𝑥-axis. This intersection point will have a

𝑦-coordinate of zero. Let’s say this intersection point

on the graph of 𝑓 inverse has the coordinates 𝑏, zero. Remember that we can swap this

ordered pair to get the corresponding point on the graph of 𝑓. The corresponding point on the

graph of 𝑓 will therefore have the coordinates of zero, 𝑏. In other words, it will be the

𝑦-axis intercept on the graph of 𝑓. Okay, so our method will then be to

find the 𝑦-axis intercept on the graph of 𝑓 and to swap the ordered pair to get

the 𝑥-axis intercept on the graph of 𝑓 inverse.

Looking at the graph of 𝑓 given in

the question, we can clearly see that the 𝑦-axis intercept has coordinates zero,

six. Swapping this ordered pair, we

conclude that the corresponding point on the graph of 𝑓 inverse will have the

coordinates six, zero. With this, we have answered our

question. The graph of 𝑓 inverse of 𝑥

intersects the 𝑥-axis at the point six, zero.

Okay, earlier in this video, we

briefly mentioned one-to-one functions and how they related to inverse

functions. Let’s explore this in more detail

now. We said that if 𝑓 is a one-to-one

function, it has an inverse function. It is also true that if 𝑓 is not a

one-to-one function, it does not have an inverse. Let us explore why this is the case

using the following graph. This function 𝑓 of 𝑥 is not a

one-to-one function. One of the ways that we can test if

something is a one-to-one function is using the horizontal line test. This says that if we can draw a

horizontal line that intersects the graph at more than one point, the graph does not

represent a one-to-one function, and we say it fails the test.

For our graph of 𝑓 of 𝑥, we can

clearly see that a horizontal line can be drawn such that it intersects the graph at

more than one point. This means that 𝑓 of 𝑥 fails the

test, and we confirm it is not a one-to-one function. Okay, we know that reflecting a

graph in the line 𝑦 equals 𝑥 gives us the graph of its inverse. Let us see what happens in this

case. Now, we could choose to do this on

the same set of axes, but let’s do this on a different set for clarity. We will tentatively call this new

graph 𝑓 inverse of 𝑥. But as we’ll see soon, this is not

the case.

Looking at this new graph, we might

notice an important detail. Remember that in order for some

relation to be called a function, each input must correspond to only one output. One way to test for this is using

the vertical line test. We’re gonna edit this statement

below to now represent the vertical line test. This test says that if we can draw

a vertical line that intersects the graph at more than one point, the graph does not

represent a function. Earlier in this video, you may

recall that we said the domain and range of some function can be said to swap for

its inverse function. If we could draw a horizontal line

that intersected our original graph of 𝑓 of 𝑥 at more than one point, it might now

be obvious that we can draw a vertical line that intersects our new graph at more

than one point. We can also clearly see this on our

diagram.

This means that what we have

tentatively called 𝑓 inverse of 𝑥 is not a function at all. Since this is not a function, we

say that our original function 𝑓 of 𝑥 does not have an inverse. Another way to think about this is

as follows. Consider our original function

𝑓. There are two values in its domain,

𝑎 one and 𝑎 two, which correspond to the same value in its range, 𝑏 one. This means that for the relation

which we supposed was the inverse of 𝑓, which we now know is not a function, the

value of 𝑏 one, which is now in its domain, must correspond to two values, 𝑎 one

and 𝑎 two, in its range.

Consider what would happen if we

try to use this relation to undo the action of our original function 𝑓 for an

output of 𝑏 one. Our supposed inverse cannot tell us

for certain whether our original input was 𝑎 one or 𝑎 two. For this reason, we say that if

something is not a one-to-one function, it does not have an inverse. Let’s take a look at an example to

illustrate this.

Determine which of the following

functions does not have an inverse.

To answer this question, we’re

going to be using the fact that if a function is not a one-to-one function, it does

not have an inverse. The way that we can test for

one-to-one functions is using the horizontal line test. This tells us that if we can draw a

horizontal line that intersects the graph at more than one point, it does not

represent a one-to-one function. And we say the graph fails the

test.

After carefully examining our

options, it should become clear that there is only one graph that we could draw a

horizontal line on so that it intersects the graph at more than one point. This is the function 𝑔 of 𝑥. We have just found that 𝑔 of 𝑥

fails the horizontal line test. This means that it’s therefore not

a one-to-one function, and so it does not have an inverse function. The answer to our question is

therefore option (b). 𝑔 of 𝑥 does not have an inverse

function, but the other three options do since they passed the horizontal line

test.

Let’s now take a look at one final

example.

By sketching graphs of the

following functions, which is the inverse of itself?

This question has directed us to

sketch these four graphs, and so this should be our first step. At this point, you should be

familiar with sketching graphs. And in order to avoid going into

unnecessary detail, this video will simply provide them. You may, however, wish to verify

these graphs yourself by drawing up a small table of values or by using graphing

software. First, we have the graph of one

over 𝑥. Next, we have the graph of 𝑥

squared, forming the familiar shape of a parabola. We then have the graph of 𝑥 cubed

and, finally, the graph of one over 𝑥 squared.

In order to move forward with this

question, we recall the following rule. Inverse functions have graphs which

are reflections of each other in the line 𝑦 equals 𝑥. This means that a graph which is

the inverse of itself has symmetry about the line 𝑦 equals 𝑥. What this means is that if a

function is the inverse of itself when reflected over the line 𝑦 equals 𝑥, the

resulting function will be the same as the original function. You may already be able to spot the

answer, but let’s reflect all of our functions in the line 𝑦 equals 𝑥 now. We first reflect one over 𝑥, then

𝑥 squared, followed by 𝑥 cubed, and finally one over 𝑥 squared.

After doing so, it should be clear

to us that the reflection of the graph one over 𝑥 is the same. It’s also the graph of one over

𝑥. The reflections of all three of the

other options are different. This means that the inverse

functions are different from the original function. Hence, 𝑥 squared, 𝑥 cubed, and

one over 𝑥 squared are not inverses of themselves. One small side note, for options

(b) and (d), the reflections are not actually functions at all. These graphs would fail the

vertical line test. Since the graphs are not functions

at all, we say that 𝑥 squared and one over 𝑥 squared do not have inverses. Okay, back to our question, it

should be clear that option (a) is the only graph for which the original and the

reflection are the same. This means that from our options

one over 𝑥 is the only function which is the inverse of itself.

To finish off this video, let’s go

through some key points. If 𝑓 is a one-to-one function, it

has an inverse function which can be represented using the following notation. We would say this as 𝑓

inverse. If 𝑓 is not a one-to-one function,

it does not have an inverse function. We can think of inverse functions

as undoing the action of each other. This can be represented as a

function composition like so. Inverse functions have graphs which

are reflections of each other in the line 𝑦 equals 𝑥. The coordinates of points which lie

on the graphs of inverse functions can be thought of as reversed ordered pairs. We saw that this can be interpreted

as the domain and the range of a function switching for its inverse. Finally, some functions are their

own inverse. The graphs of these functions have

reflectional symmetry in the line 𝑦 equals 𝑥.