Inverse functions are functions that are symmetric to the y = x line. Another way of thinking about them is that the x’s and y’s are swapped. For example if the original function had the point (-2, 5) on its graph, then (5, -2) would be on its inverse. Usually one is asked to find the inverse of a function algebraically. To do that, follow these steps.

1)      Switch x and y.

2)      Solve for y.

 

 

Example

 

Find the inverse of f(x)= 2 x + 3

 

Before doing step 1, one has to have a y. So rewrite the function with a y. Remember by the definition of functions that f(x) = y.

 

y = 2x + 3

 

Using step 1, swap x and y.

 

x = 2y + 3

 

Now solve for y by first subtracting 3 from both sides.

 

x-3 = 2y

 

Divide both sides by 2.

 

 

Rewrite the answer by flipping it around.

 

 

When writing the inverse of a function, use the notation f ^-1(x).

 

 

Answer!

 

Note- After finding the inverse, graph the result and apply the vertical line test to it to see if it is a function. In this case, it is a function so write the answer. If it turns out not to be a function, simply write “does not exist” as an answer.

 

 

 

 

 

Example

 

Find the inverse, if it exists, of y = 2x² + 1

 

First, swap x and y.

 

x = 2y² + 1

 

Solve for y by first subtracting 1 from each side.

 

x - 1 = 2y²

 

Divide both sides by 2.

 

 

Before taking the square root of both sides, flip the equation around so it is easier to understand.

 

 

Take the square root of both sides.

 

y = ±

 

Before writing this as the answer, graph it and see if it is a function. When doing so, realize that the inverse is not a function.

 

Does not exist

 

Answer!

 

Here is the graph of the previous problem, notice that it would not pass the vertical line test.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example

Find the inverse, if it exists, of y =

 

Swap x and y.

 

Solve for y by first multiplying both sides by y.

 

y x = 3y – 7

 

Subtract 3y by both sides.

 

yx – 3 y = - 7

 

Factor out a y on the left side.

y(x-3) = -7

 

Divide both sides by x-3.

 

 

Graphing this one would realize that this is a function so write it using f ^-1(x).

 

 

Answer!

 

In the previous example the result would have a limited domain. In other words, x cannot equal -3. This may not need to be stated in a problem but if it is, just write beside the answer that the domain is all real numbers ¹ -3.