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## Algebra 1

### Unit 8: Lesson 13

Intro to inverse functions

# Intro to inverse functions

Sal explains what inverse functions are. Then he explains how to algebraically find the inverse of a function and looks at the graphical relationship between inverse functions. Created by Sal Khan.

## Want to join the conversation?

• Is the inverse of y=4 x=4? If so then are horizontal and vertical lines the only lines that are perpendicular to there inverses? •   I love this question-- because testing the boundaries of new concepts is so important to advancing mathematics.First, we must come to grips with the idea that not every function has an inverse. Only functions with "one-to-one" mapping have inverses.The function y=4 maps infinity to 4. It is a great example of not a one-to-one mapping. Thus, it has no inverse. There is no magic box that inverts y=4 such that we can give it a 4 and get out one and only one value for x.
• •  Justin,

If you are trying to invert a function, one way to do it is to switch the positions of all of the variables, and resolve the function for y. The intuition works like this:

We sometimes think about functions as an input and an output. For example, we take a value, called x, and that is what we put into the function. Then the function does some "stuff" and we get out a value called y. So, for some function f, X goes in, and Y comes out. If we think about it that way, then for the inverse of the f function (call it 'g', maybe), we should be able put IN the values that came OUT of function f as our y's, and get the same x values we put IN to f to get the y's originally.

But that is kind of like we switched the x's and y's in our f function…. and that's exactly how you solve for the inverse function, g. You take the original function, switch all of the y's for x's and the x's for y's, and then you resolve it for y.

For example: if our original function f is y=2x-5, then we would switch the y's and x's to get x=2y-5. If we solve for y, we get y=(x+5)/2. That's function 'g'.

Now let's try it out. We put in an x=0, 1, and 2 in function f, and we get, -5,
-3, and -1 as the corresponding y's (try it yourself). Now we take those y's and we make them our x values (or inputs) into function g and we should get our original 0, 1, and 2.
y=(-5+5)/2 =0
y=(-3+5)/2= 1
y=(-1+5)/2=2

It worked. The x's (or inputs) for our first function produce y's (outputs) from our first function. We can take those y's (outputs from our first function) and make those the x's (or inputs) of our inverse function, and we get the original inputs we started with.
• i don't quite get the thing Sal does at . He is talking about y being equal to x, and then draws a dotted line in the middle. is there maybe a video that clarifies this relation? • At around , Sal labels the function 1/2y-2 as an inverse function, deemed f-1(y).

I am not sure, but I think this is incorrect? F-1(x) should equal 1/2y-2, not F-1(y), right? F(y) is just a function, not an inverse. However, the inverse of x is equal to the y function.

I hope I wasn't too confusing, but it would be appreciated if this got cleared up! :) thanks • f(x)= 2x +4 .... f^-1(x)=(x/2)-2 .... f^-1[f(x)]=(2x+4)/2 -2 = x .(identity)
f(y)=(y/2)-2.... f^-1(y)= 2y+4.... f^-1[f(y)]=2(y/2 -2)+4 = y .(identity)
Check plot at to see that these are symmetric across X=Y. ..Venn rings can mislead.
• as to solve for x, for inverse, why can't we just switch the x and y in the same equation..? • • No, all strictly growing or strictly decreasing functions have an inverse.

If it is not strictly growing/decreasing, there will be values of f(x) where
f(x) = f(y), x not equal to y.

So, its inverse g would have two values for f(x), as g( f(x) ) = x AND y, which is not possible for a function.

An example of this is x^2. It's inverse would be g(x) = +sqrt( x ) AND -sqrt( x ), which is not possible.

However, functions such as f( x ) = x^3, or f( x ) = e^x, which are strictly growing, do have an inverse : )
• Starting at Sal draws another set as the range. Why is Sal drawing a second set next to the first one if the domain and the range are the same set, the set of real numbers? Shouldn't he draw one "blob" such that the function maps from elements in that set to other elements in that set? • No, because the domain is the numbers on x that can go in and the range is the numbers on y that can be a result of this function. So they could both be all real numbers but they could be different. In a square root function for example, the domain would be all positive numbers only.
• Can a function be the inverse of itself?
What are some examples? • There are actually other ways you can write an inverse function! Take this example:

f(x) = 2x + 4
f(x)^-1 = (x-4)/2  