Main content

### Course: Precalculus > Unit 1

Lesson 4: Inverse functions in graphs and tables# Reading inverse values from a graph

Learn how to find the inverse of a function and how to graph it in this video. The inverse function swaps the x and y values of the original function, and the graph of this inverse function is a reflection of the original function about the line y=x. Created by Sal Khan.

## Want to join the conversation?

- where is b coming from?(2 votes)
- The a and b are two example variable values that Sal uses. If you have a point (a, b), where the a and b were any random numbers, this means that when you put "a" into the function, you get "b" back. Sal then says that this means that when you put b into the inverse of the function, you should get a back. By choosing points on the graph as (a, b), we can find the corresponding points on the graph of the inverse as (b, a), and then fill in the inverse's graph that way.(6 votes)

- I get this mostly, reflecting across the diagonal axis, switching x and y, etc., but how can f^-1(2) be x=4 when that point clearly doesn't fall within y = f^-1(x)? Does f^-1(x) always return an x value in f(x), and y = f^-1(x) return the inverse function?(3 votes)
- Inverting a function does not invert the actual values.

If f(x)=2, but the graph intercepts y=2 at x=4. Hope this helped!(1 vote)

- I would have graphed the x-intercept as the y-intercept to have a more accurate sketch because the x-intercept of f appeared to be (7,0) so the y-int of f inverse would be (0,7)(2 votes)
- wait i thought f-1(2) is 1/2 isnt it the inverse of 2 the reciprocal so 1/2? how did he get 4(1 vote)
- With functions, inverting the function does not mean inverting the value, but inverting the x and y values of the function. Normally, f(2)=3.5 because when x=2, then y=3.5 according to the equation of the function.

When a function is inverted, however (on a graph at least), we would look at the y value of the original function and find what the value of x is when y is that value, in this case, 2.

So, on the function, where y=2, x=4. Hope this helps.(2 votes)

- he never explained what he means by "looking like a reflection about the line ..." what does that mean? it's not a mirror image, so what other kinds of reflections are there? is there a video that talks about how to detect reflection about the line ...?(1 vote)
- its a reflection of the line y=x.

if you draw a line y=x you will see that the two graphs are reflected accross that line. there is probably a video on graph reflections too.(2 votes)

- Is alg 2 and pre-cal the same?(1 vote)
- No, but a lot of Alg goes into Pre Cal(1 vote)

- How do you find the corresponding x(1 vote)
- Why couldn't we just say f inverse of x means you use the y-value instead of the x, as it "Inverses" the function, which we've already learnt?

Is there some actual mathematical reason, because I didn't understand what the video showed 😅(1 vote)- Is there some actual mathematical reason

So is there statement or something Sal said that you didn't understand.(1 vote)

## Video transcript

- [Instructor] We're told
the following graph shows y is equal to f of x, all right. And then the first question they say is, "What appears to be the
value of f inverse of two?" Pause the video and see if
you can have a go at that. All right, now let's work
through this together. So it's important to realize they're not asking us what f of two is. They're asking us f inverse of two. f of two we would say,
"Okay, when x is equal to two that's the input into our function." And then the graph tells us that f of two it looks like it might be
a little bit more than 2.5. Maybe it's approximately 2.6. But that's not what they're asking us. They're asking us f inverse of two. And just as a reminder of
what an inverse function is, if we have some input, x, and we input into our function, f, that is going to output f of x. Now, if we were to input f of x into the inverse function
for f, then the output here, which is going to be f inverse of f of x is going to get us back
to this original x, is going to be equal to x. So we are really, in this scenario, dealing with this part of this
chain of inputs and outputs. We're saying we want to figure
out what f inverse of two is. So this part, right over here
is going to be equal to two. So we're saying when f
of x is equal to two, what is the corresponding x? So when f of x is equal to two,
what is the corresponding X? We get four. So let's write that down. f inverse of two is equal to four. So when x is equal to four, you input that f of four is equal to two or f inverse of two is equal to four. Now the next part they say, "Sketch the graph of y is
equal to f inverse of x." So an important thing to realize is if we're saying that
b is equal to f of a which implies that the point
a,b is on the graph of f, then we're dealing with f inverse. We would know that a would
be equal to f inverse of b. You can think about swapping these two, the a and the b's here, which means that b,a
would be on f inverse. So any coordinate point that's
on our original graph f, if you swap the x and the y, that's going to be on our f inverse. So let's just pick some points. And once again, they're
just saying sketch, so it doesn't have to be perfect. So if we look at this
point right over here, that looks like the point
-10, maybe it looks like 3.4. Well, then that means
if we swap that x and y it'll be on the graph of f inverse. So if we go to 3.4, and then -10, so it gets us right about there, that would be on the graph of f inverse. Now, if we went to, let's say let's say this point right over here. This is the point -2,3. So if -2,3 is on the graph of f, then 3,-2 would be on the inverse. So 3,-2 would be on the
inverse right over there. Let's pick a few more points. So we have this point right over here, 4,2 which would be on the graph of f, which means that 2,4 is on the inverse. So 2,4, which would be right over there. And then if we look at
this point over here which looks like roughly nine, let's just call it 9,-10 maybe it's 9.1,-10. If that's on the graph of f, then if we swap that -10, maybe 9.1 would be on the inverse
-10,9.1, right over there. And so then we could connect the dots to try to sketch out
what the inverse function is going to look like, the
graph of the inverse function. So it's going to look something like that. And you might notice, it looks like it's a reflection about the line y=x. It looks like it's a
reflection about that line, which is exactly right.