## What Is an Inverse Function?

**The inverse function **$f^{-1}$** undoes the action performed by the function f. We read **$f^{-1}$** as “f inverse.”**

If $f^{-1}$ is an inverse of the function f, then f is an inverse function of $f^{-1}$. Thus, we can say that f and $f^{-1}$ reverse each other.

A function from set A to the set B, represented by $f:A \rightarrow B$ is a relation from the set A (a set of inputs) to the set B (a set of possible outputs) such that every element in A is related to exactly one element from the set B.

**Domain: **The set of inputs/ the set of values for which the function is defined/ the set of values that we can plug into the function

**Range:** The set of outputs/ the set of resulting values

The symbol for inverse function is $f^{-}1\; (f$ with the exponent of $\;﹣\;1)$.

## Inverse Function Definition in Math

A function $g = f^{-1}$ is said to be an inverse function of a function $y = f(x)$ if whenever $f(x) = y$, we have $g(y) = f^{-1}(y) = x$. If f and g are inverse functions, then we have $f(x) = y$ if and only if $g(y) = x$.

## Inverse Function Formula

If f maps x to $f(x)$, then $f^{-1}$ maps f(x) back to x. Thus, if you apply a function and then its inverse, you get the original value. Same applies if we reverse the order of the two functions. Thus, the inverse function formula can be given as

$f^{-1}(f(x)) = x$** **…for all x in the domain of f

and

$f(f^{-1}(y))=y$ …for all y in the domain of $f^{-1}$

**If domain **$=$** range, we can say that**

$f^{-1}(f(x)) = f(f^{-1}(x)) = x$

## $f^{-1}$ Meaning and Mapping Diagram

Consider a mapping diagram of a function f and its inverse $f^{-1}$ .

What do you notice?

The domain of f is the range of $f^{-1}$.

The domain of $f^{-1}$ is the range of f.

Action of the function f | Action of the inverse function $f^{-1}$ |
---|---|

f takes 1 to a. | $f^{-1}$ takes a to 1. |

f takes 2 to b. | $f^{-1}$ takes b to 2. |

f takes 3 to c. | $f^{-1}$ takes c to 3. |

f takes 3 to d. | $f^{-1}$ takes d to 4. |

## How to Find the Inverse of a Function

Let’s understand the steps to find the inverse of a function with an example.

Let us consider a function $f(x) = ax + b$.

**Step 1: For the given function, replace **$f(x)$** by y. In other words, substitute **$f(x) = y$**.**

Put $f(x) = y$ in $f(x) = ax + b$.

The result is $y = ax + b$.

**Step 2: Replace ****x**** with ****y****. Replace ****y**** with ****x****. **

In the function $y = ax + b$, we interchange x and y.

The result is $x = ay + b$.

**Step 3: Solve the expression to write it in terms of **y**.**

$x = ay + b \Rightarrow y = $\frac{x \;-\; b}{a}$

The result is $y = \frac{x \;-\; b}{a}$

**Step 4: Replace **$y = f^{-1}(x)$**.**

The result is $f^{-1}(x) = \frac{x \;-\; ba}$.

Thus, $f ^{-1}(x)= \frac{x \;-\; b}{a} is an inverse function of $y = f(x) = ax + b$.

## Inverses of Some Common Functions

Some common inverse functions are given below:

Function | Inverse Function | Conditions |
---|---|---|

$ax + b$ | $x \;-\; ba$ | $a \neq 0$ |

$\frac{1}{x}$ | $\frac{1}{y}$ | x and y not equal to 0 |

$x^2$ | $\sqrt{y}$ | $x$ and $y ≥ 0$ |

$e^x$ | ln(y) | $y > 0$ |

sin(x) | $sin^{-1}(y)$ | $\frac{-\pi}{2}$ to $\frac{-\pi}{2}$ |

cos(x) | $cos^{-1}(y)$ | 0 to |

tan(x) | $tan^{-1}(y)$ | $\frac{\pi-}{2}$ to $\frac{-\pi}{2}$ |

## Does Every Function Have an Inverse?

Not every function has an inverse. A function has an inverse if and only if it is one-to-one (or bijective). A function f has an inverse function if and only if, for every element y in its range, there is only one value of x in its domain for which $f(x) = y$.

It means that for each output, there is precisely only one input. The function does not take the same value twice. In other words, a function f is one-to-one if, for every a and b in its domain, $f(a) = f(b)$ implies $a = b$.

## Inverse Function Graph

**The graphs of f and **$f\;-\;1$**are symmetric over the line **$x = y$**. **The functions f and $f\;-\;1$ are mirror images of each other on a graph since the roles of these two variables are reversed. Thus, we can identify whether two functions are inverses of each other simply by checking if they are symmetric over $x = y$.

## What Are the Types of Inverse Functions?

The different types of inverse functions include inverse trigonometric functions, inverse of rational functions, inverse hyperbolic functions, and inverse of logarithmic functions.

## Inverse of Rational Functions

A rational function is an algebraic function such that both numerators and denominators are polynomials. It is a function of the form $f(x) = \frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$.

The steps to find the inverse of a rational function are:

**Step 1:** Substitute $f(x) = y$.

**Step 2:** Interchange x and y.

**Step 3:** Solve for y and express it in terms of x.

**Step 4:** Replace y with $f^{-1}(x)$.

The final result we get is the inverse function.

**Example: Find the inverse of **$f(x) = \frac{x + 1}{5 \;-\; 2x}. (x \neq \frac{5}{2}$ and $x \neq \frac{-1}{2})$**.**

Here, $y = \frac{x + 1}{5 \;-\; 2x}$ … $x\neq \frac{5}{2}$

Interchange x and y.

$x = \frac{y + 1}{5 \;-\; 2y}$

$5x\;-\;2xy = y + 1$

$5x\;-\;1 = 2xy + y$

$5x\;-\;1 = y(2x + 1)$

$y = \frac{5x \;-\; 1}{2x + 1}$

$f^{-1}(x) = \frac{5x \;-\; 1}{2x + 1}$ … $x \neq \frac{-1}{2}$

## How to Find Inverse Functions Using Algebra

We can replace f(x) with y and solve the algebraic expression in terms of x.

Consider an example.

Function: $f(x) = 5x \;-\; 9$

Replace the f(x) with y: $y = 5x \;-\; 9$

Add 9 to both sides: $y + 9 = 5x$

Divide both sides of the equation by 5: $\frac{y + 9}{5} = x$

Exchange the sides: $x = \frac{y + 9}{5}$

Solve the expression replacing x with $f^{-1}(y) = \frac{y + 9}{5}$

## Facts about Inverse Function

- The inverse of f(x) is $f^{-1}(y)$.

Inverse function $f^{-1}(x)$ is not the same as the reciprocal $\frac{1}{f(x)}$.

**Horizontal line test:**If a horizontal line intersects the graph of a function at more than two points, the function does not have an inverse. If the horizontal line intersects the graph only at a single point, the graph is one-to-one.

- Inverse Logarithmic Functions and Inverse Exponential Function: The natural log functions are inverse of the exponential functions.

- If f and g are inverse functions of each other, then $f(g(x)) = g(f(x) = x$
- Sometimes we have to restrict the domain of a function so that the inverse function can be defined. For example, the function $f(x) = x^2\;-\;5$ does not have an inverse if the domain is the set of real numbers. However, if the domain is restricted to $x \leq \;-\;5$, the function becomes one-to-one and has an inverse.
- The natural logarithmic function is the inverse of the exponential function. The inverse of the exponential function $y = ax$ is $x = a^y$. Note that the logarithmic function $y = log_a x$ is equivalent to the exponential equation $x = a^y$.

## Conclusion

In this article, we learnt about Inverse functions, their graphs, and steps for finding inverse functions. Let’s solve a few solved examples and practice problems.

## Solved Examples On Inverse Function

**1. What is the inverse of the function **$f(x) = x + 1$**?**

**Solution:**

**Given function: **$f(x) = x + 1$

Replace f(x) by y.

$y = x + 1$

Interchange x and y.

** **$x = y + 1$

Solve for y.

$y = x\;-\;1$

Replace** **y by $f^{-1}(x)$.

$f^{-1}(x) = x\;-\;1$

**2. Find the inverse of **$f(x) = x$**?**

**Solution: **

Given function: $f(x) = x$

Replace f(x) by y.

$y = x$

Interchange x and y.

$x = y$

Solve for y.

$y = x$

Replace** **y by $f^{-1}(x)$.

$f^{-1}(x) = x$

The inverse of an identity function is the identity function itself.

**3. What is the inverse of a function **$g(x) = 5(x + 3)$**?**

**Solution: **

$g(x) = 5(x + 3)$

$y = 5(x + 3)$

$x = 5(y + 3)$

$x = 5y + 15$

$y = \frac{x \;-\; 15}{5}$

$g\;-\;1(x) = \frac{x \;-\; 15}{5}$

**4. If **$g(x) = 8\;-\;\frac{x}{3}$**, and **$f(x) = 24\;-\;3x$** , show that f and g are inverses.**

**Solution:**

**i) **$g(x) = 8\;-\;\frac{x}{3}$

$f(g(x)) = f(8\;-\;\frac{x}{3})$

$f(g(x)) = 24\;-\;3(8\;-\;\frac{x}{3})$

$f(g(x))= 24\;-\;24 + x$

$f(g(x)) = x$

**ii) **$f(x) = 24\;-\;3x$

$g(f(x)) = g(24\;-\;3x)$

$g(f(x)) = 8\;-\;\frac{(24\;-\;3x)}{3}$

$g(f(x)) = 8\;-\;8 + x$

$g(f(x)) = x$

From (i) and (ii), we have $f(g(x)) = x = g(f(x))$

Thus, f and g are inverses.

## Practice Problems On Inverse Function

## Inverse Function - Definition, Types, Examples, Facts, FAQs

### $f^{-1}(f(x))=$

$f^{-1}(f(x)) = x$

### A function has an inverse if and only if it is

A function has an inverse if and only if it is bijective (one-to-one).

### If f and g are inverse functions, then we have $f(x) = y$ if and only if

If f and g are inverse functions, then we have $f(x) = y$ if and only if $g(y) = x$.

### The graphs of a function and its inverse are symmetric over

The graph of a function and its inverse are symmetric over the line $x = y$.

### What is the inverse of $f(x) = x + 5$?

$f(x) = x + 5$

$y = x + 5$

$x = y + 5$

$y = x \;-\; 5$

Thus, $f^{-1}(x) = x\;-\;5$

## Frequently Asked Questions On Inverse Function

**What are inverse operations?**

Inverse operations are opposite operations that reverse or cancel the action of one another.

Addition and subtraction are inverse operations.Multiplication and division are inverse operations.

**What is a bijective function?**

Bijective function is a one-one and onto function.

**Is **$f^{-1}(x)$** same as **$f(x)^{-1}$**?**

No. $f^{-1}(x)$ is the inverse of the function f.

$f(x)^{-1} = \frac{1}{f(x)} =$ Reciprocal of $f(x)$