How do we find the inverse of a function?

It has been easy so far, because we know the inverse of Multiply is Divide, and the inverse of Add is Subtract, but what about other functions?

Table of Contents Show

Example: Square and Square Root
But we can fix that!
Example: (continued)
No Inverse?
Domain and Range
Example: Square and Square Root (continued)
How do you find the inverse of a function?
Why do we find inverse functions?

Here is a list to help you:

InversesCareful!

<=>

Don't divide by zero1x<=>1yx and y not zerox2<=>x and y ≥ 0xn<=>orn not zero
(different rules when n is odd, even, negative or positive)ex<=>ln(y)y > 0ax<=>loga(y)y and a > 0sin(x)<=>sin-1(y)cos(x)<=>cos-1(y)tan(x)<=>tan-1(y)

(Note: you can read more about Inverse Sine, Cosine and Tangent.)

Careful!

Did you see the "Careful!" column above? That is because some inverses work only with certain values.

Example: Square and Square Root

When we square a negative number, and then do the inverse, this happens:

Inverse (Square Root): √(4) = 2

But we didn't get the original value back! We got 2 instead of −2. Our fault for not being careful!

So the square function (as it stands) does not have an inverse

But we can fix that!

Restrict the Domain (the values that can go into a function).

Example: (continued)

Just make sure we don't use negative numbers.

In other words, restrict it to x ≥ 0 and then we can have an inverse.

So we have this situation:

x2 does not have an inverse
but {x2 | x ≥ 0 } (which says "x squared such that x is greater than or equal to zero" using set-builder notation) does have an inverse.

No Inverse?

Let us see graphically what is going on here:

To be able to have an inverse we need unique values.

Just think ... if there are two or more x-values for one y-value, how do we know which one to choose when going back?

General FunctionNo Inverse

Imagine we came from x1 to a particular y value, where do we go back to? x1 or x2?

In that case we can't have an inverse.

But if we can have exactly one x for every y we can have an inverse.

It is called a "one-to-one correspondence" or Bijective, like this

Bijective FunctionHas an Inverse

A function has to be "Bijective" to have an inverse.

So a bijective function follows stricter rules than a general function, which allows us to have an inverse.

Domain and Range

So what is all this talk about "Restricting the Domain"?

In its simplest form the domain is all the values that go into a function (and the range is all the values that come out).

As it stands the function above does not have an inverse, because some y-values will have more than one x-value.

But we could restrict the domain so there is a unique x for every y ...

... and now we can have an inverse:

Note also:

The function f(x) goes from the domain to the range,
The inverse function f-1(y) goes from the range back to the domain.

Let's plot them both in terms of x ... so it is now f-1(x), not f-1(y):

f(x) and f-1(x) are like mirror images
(flipped about the diagonal).

In other words:

The graph of f(x) and f-1(x) are symmetric across the line y=x

Example: Square and Square Root (continued)

First, we restrict the Domain to x ≥ 0:

{x2 | x ≥ 0 } "x squared such that x is greater than or equal to zero"
{√x | x ≥ 0 } "square root of x such that x is greater than or equal to zero"

And you can see they are "mirror images"
of each other about the diagonal y=x.

Note: when we restrict the domain to x ≤ 0 (less than or equal to 0) the inverse is then f-1(x) = −√x:

To find the inverse of a function, start by switching the x's and y's. Then, simply solve the equation for the new y. For example, if you started with the function f(x) = (4x+3)/(2x+5), first you'd switch the x's and y's and get x = (4y+3)/(2y+5). Then, you'd solve for y and get (3-5x)/(2x-4), which is the inverse of the function. To learn how to determine if a function even has an inverse, read on!

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How do you find the inverse of a function?

How do you find the inverse of a function? To find the inverse of a function, write the function y as a function of x i.e. y = f(x) and then solve for x as a function of y.

Why do we find inverse functions?

An inverse function essentially undoes the effects of the original function. If f(x) says to multiply by 2 and then add 1, then the inverse f(x) will say to subtract 1 and then divide by 2. If you want to think about this graphically, f(x) and its inverse function will be reflections across the line y = x.