How to Find the Inverse of a Function

Co-authored by Mario Banuelos, PhD

Last Updated: November 29, 2024 Fact Checked

A foundational part of learning algebra is learning how to find the inverse of a function, or f(x). The inverse of a function is denoted by f^-1(x), and it's visually represented as the original function reflected over the line y=x. This article will show you how to find the inverse of a function.

Steps

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1
Make sure your function is one-to-one. Only one-to-one functions have inverses.^{[1]
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- A function is one-to-one if it passes the vertical line test and the horizontal line test. Draw a vertical line through the entire graph of the function and count the number of times that the line hits the function.^{[2]
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  Research source} Then draw a horizontal line through the entire graph of the function and count the number of times this line hits the function. If each line only hits the function once, the function is one-to-one.
  - If a graph does not pass the vertical line test, it is not a function.
- To algebraically determine whether the function is one-to-one, plug in f(a) and f(b) into your function and see whether a = b. As an example, let's take f(x) = 3x+5.
  - f(a) = 3a + 5; f(b) = 3b + 5
  - 3a + 5 = 3b + 5
  - 3a = 3b
  - a = b
- Thus, f(x) is one-to-one.
2
Given a function, switch the x's and the y's. Remember that f(x) is a substitute for "y."^{[3]
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- In a function, "f(x)" or "y" represents the output and "x" represents the input. To find the inverse of a function, you switch the inputs and the outputs.
- Example: Let's take f(x) = (4x+3)/(2x+5) -- which is one-to-one. Switching the x's and y's, we get x = (4y + 3)/(2y + 5).
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3
Solve for the new "y." You'll need to manipulate the expressions to solve for y, or to find the new operations that must be performed on the input to obtain the inverse as an output.^{[4]
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- This can be tricky depending on your expression. You may need to use algebraic tricks like cross-multiplication or factoring to evaluate the expression and simplify it.
- In our example, we'll take the following steps to isolate y:
  - We're starting with x = (4y + 3)/(2y + 5)
  - x(2y + 5) = 4y + 3 -- Multiply both sides by (2y + 5)
  - 2xy + 5x = 4y + 3 -- Distribute the x's
  - 2xy - 4y = 3 - 5x -- Get all the y terms on one side
  - y(2x - 4) = 3 - 5x -- Reverse distribute to consolidate the y terms
  - y = (3 - 5x)/(2x - 4) -- Divide to get your answer
4
Replace the new "y" with f^-1(x). This is the equation for the inverse of your original function.^{[5]
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- Our final answer is f^-1(x) = (3 - 5x)/(2x - 4). This is the inverse of f(x) = (4x+3)/(2x+5).

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Question

Where did the +5 in the determining whether the function is one-to-one go?

Cluster Duck

Community Answer

The 5's cancel each other out during the process. Here is the extended working out. 3a + 5 = 3b + 5, 3a +5 -5 = 3b, 3a = 3b.

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Question

f(x+3)=3x+5 , find inverse

I_l1ke_gam3s

Community Answer

First convert from f(x+3) into the form f(x), call it g(x) = 3(x - 3) + 5 = 3x - 9 + 5 = 3x - 4. Let g(x) = y. Then y = 3x - 4. Switch around: x = 3y - 4. x + 4 = 3y, so y = (x + 4)/3. So the inverse function is f⁻¹(x) = (x + 4)/3

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Question

3x²+5x / 2x+4x². Find the inverse function.

BullyHelper

Community Answer

Set this equal to f(x). Then replace this with y. Then switch the two letters around. x = 3y^2 + 5y/2y + 4y^2. Then multiply both sides by the denominator. x(2y + 4y^2) = 3y + 2. 2y + 4y^2 = 3y + 2 / x. Then 2y = 3y + 2 / x - 4y^2. and y = 3y + 2 / x - 4y^2 / 2.

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