What Are Inverse Functions?
Functions are relationships between mathematical values.
Functions have input values, and an operation on each input value results in a single output value.
For some algebra problems, we want to work backwards from the output to discover what the input was.
When we work backwards to find the input, we use what is called an inverse function.
If you need to review functions before completing these exercises, see our post on Function Notation.
You can also try our Algebra Practice Test.
Inverse Functions – Example
An example of a function that has an inverse is:
ƒ(x) = 2x + 3.
The inverse of this function is written as follows:
f–1(x) = (x – 3) ÷ 2
In the notation for the inverse function above, you will notice that the –1 exponent is given after the function.
The –1 exponent is a special notation used to indicate an inverse function.
Setting Up Inverse Functions
Let’s look at our functions again.
Our original function was:
ƒ(x) = 2x + 3
Our inverse function was:
f–1(x) = (x – 3) ÷ 2
To set up an inverse function, you usually need to perform four steps.
Step 1:
Replace ƒ(x) with y.
ƒ(x) = 2x + 3
y = 2x + 3
Step 2:
Then isolate x to one side of the equation.
y = 2x + 3
y – 3 = 2x + 3 – 3
y – 3 = 2x
(y – 3) ÷ 2 = x
x = (y – 3) ÷ 2
Step 3:
Then swap x and y.
x = (y – 3) ÷ 2
y = (x – 3) ÷ 2
Step 4:
Finally, replace y with the function f–1(x) to solve.
y = (x – 3) ÷ 2
f–1(x) = (x – 3) ÷ 2
Inverse Functions – Proofs
To prove an inverse function, put a value for x into the original function.
Let’s use x = 6 in our original function as an example.
ƒ(x) = 2x + 3
ƒ(x) = (2 × 6) + 3
ƒ(x) = 12 + 3
ƒ(x) = 15
Then take the result, or the output, from the original function and use it as in input in the inverse function.
f–1(x) = (x – 3) ÷ 2
f–1(x) = (15 – 3) ÷ 2
f–1(x) = 12 ÷ 2
f–1(x) = 6
So, our result or output from the inverse function is the same as our input in the original function.
Functions Without Inverses
Not every function have as inverse.
This is the case when two or more inputs to the function result in the same output.
For instance, consider the following function: ƒ(x) = x2
When x = 2, the output is 4, but when x = –2, the output is also 4.
Therefore, the function ƒ(x) = x2 does not have an inverse.
A function will not have an inverse if any of the following are true:
- The inverse function has an even-numbered exponent.
- The inverse function has an absolute value symbol.
- The graph of the function is a flat line.
Inverse Functions – Exercises
Find the inverses for the following functions.
If the function does not have an inverse, provide an explanation.
- ƒ(x) = 5x + 4
- ƒ(x) = 2x – 7
- ƒ(x) = x4
- ƒ(x) = (x + 2) ÷ (x – 3)
- ƒ(x) = | x – 3 |
Inverse Functions – Answers
Answer 1:
ƒ(x) = 5x + 4
Replace ƒ(x) with y.
y = 5x + 4
Then isolate x to one side of the equation.
y = 5x + 4
y – 4 = 5x + 4 – 4
y – 4 = 5x
(y – 4) ÷ 5 = x
x = (y – 4) ÷ 5
Then swap x and y.
x = (y – 4) ÷ 5
y = (x – 4) ÷ 5
Finally, replace y with the function f–1(x) to solve.
y = (x – 4) ÷ 5
ƒ-1(x) = (x – 4) ÷ 5
Answer 2:
ƒ(x) = 2x – 7
Replace ƒ(x) with y.
y = 2x – 7
Then isolate x to one side of the equation.
y = 2x – 7
y + 7 = 2x – 7 + 7
y + 7 = 2x
(y + 7) ÷ 2 = x
x = (y + 7) ÷ 2
Then swap x and y.
x = (y + 7) ÷ 2
y = (x + 7) ÷ 2
Finally, replace y with the function f–1(x) to solve.
y = (x + 7) ÷ 2
ƒ-1(x) = (x + 7) ÷ 2
Answer 3:
ƒ(x) = x4
This function does not have an inverse because it has an even numbered exponent.
In other words, it is not a one-to-one function because there is more than one input for a single output.
For instance, the inputs –2 and 2 both result in the output of 16.
Answer 4:
ƒ(x) = (x + 2) ÷ (x – 3)
Replace ƒ(x) with y.
y = (x + 2) ÷ (x – 3)
Then isolate x to one side of the equation.
y = (x + 2) ÷ (x – 3)
y × (x – 3) = (x + 2)
xy – 3y = x + 2
xy – 3y – x = 2
xy – 3y + 3y – x = 2 + 3y
xy – x = 2 + 3y
x(y – 1) = 2 + 3y
x = (2 + 3y) ÷ (y – 1)
Then swap x and y.
x = (2 + 3y) ÷ (y – 1)
y = (2 + 3x) ÷ (x – 1)
y = (3x + 2) ÷ (x – 1)
Replace y with the function f–1(x) to solve.
y = (3x + 2) ÷ (x – 1)
ƒ-1(x) = (3x + 2) ÷ (x – 1)
Answer 5:
ƒ(x) = | x – 3 |
This function does not have an inverse because it has the absolute value symbol.
Like question 3 above, it is not a one-to-one function because there is more than one input for a single output.
For instance, the inputs x = 1 and x = 5 both result in the output of 2.
| 1 – 3 | = | –2 | = 2
| 5 – 3 | = | 2 | = 2
More Practice with Inverse Functions
If you are testing in trigonometry, you should also look at our practice problems on trig functions.