Imagine a mathematical conveyor belt. You feed an input into a function, the function performs a set of operations, and it spits out an output. Now, imagine you want to run the conveyor belt in reverse—to take that output, feed it backwards through the system, and recover your original input.

This "undo" mechanism is exactly what an inverse function does. It acts as the mathematical reverse gear, completely undoing the operations of the original function.

In the real world, inverse functions are everywhere. If you have a function that converts temperature from Celsius to Fahrenheit, its inverse function is the formula that converts Fahrenheit back to Celsius. In cryptography, if a function encrypts a private password into a secure string of code, the inverse function is the decryption key that recovers the original password.

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Core Principles of Inverse Functions

To write mathematical definitions for exams, we use the superscript notation $f^{-1}(x)$ to represent the inverse of a function $f(x)$ .

Key Points

The superscript " $-1$ " in $f^ -1(x)$ is NOT an exponent.
It does NOT mean $1 / f(x)$ .
$f^-1(x)$ represents the inverse function (the undo rule).
$[f(x)]^-1$ represents the reciprocal, which is 1 / f(x).

The Domain and Range Swap

Because an inverse function reverses inputs and outputs:

The domain (allowable inputs) of $f^{-1}(x)$ is exactly equal to the range (possible outputs) of the original function $f(x)$ .
The range of $f^{-1}(x)$ is exactly equal to the domain of $f(x)$ .

The Identity Composition Rule

If you apply a function and then immediately apply its inverse (or vice versa), they cancel each other out completely, leaving you with your original input $x$ :

f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x

Graphical Symmetry

If you plot $f(x)$ and $f^{-1}(x)$ on the same coordinate grid, their curves are perfectly symmetrical to each other. One is a direct reflection of the other across the diagonal bisector line $y = x$ . This is because the process of finding an inverse swaps every coordinate $(a, b)$ on the original curve to $(b, a)$ on the inverse curve.

Graph illustrating the inverse of a function as the refliection across a diagonal bisector — Image Source: Gianpiero Placidi

The One-to-One (Bijective) Requirement

Before you can calculate the inverse of a function, you must first check if an inverse is mathematically allowed to exist.

An inverse function can only exist if the original function is one-to-one (injective). This means that for every unique output y, there is exactly one unique input x.

If a function is many-to-one (for example, the quadratic curve $f(x) = x^2$ , where both $x = 2$ and $x = 2$ give the output $y = 4$ , a true inverse cannot exist over its entire domain. If you tried to inverse it, the input $y = 4$ would point to two different outputs ( $2$ and $-2$ ), violating the fundamental definition of a function.

To find the inverse of a many-to-one function, you must first restrict its domain (for example, restricting $f(x) = x^2$ to only positive values, $x \ge 0$ to make it one-to-one.

Step-by-Step Method to Calculate the Inverse

To find the algebraic rule for $f^{-1}(x)$ , follow this reliable four-step method:

Step	Action	Example: f(x) = 2x + 3
1	Replace f(x) with y	$\text{[math]}$
2	Rearrange the equation to isolate x	$\text{[math]}$
3	Swap the variables: replace x with $\text{[math]}$ and y with x	$\text{[math]}$
4	Verify your inverse using composition (Optional)	$\text{[math]}$

Exam-Style Worked Examples

Let's look at three standard exam-style problems, starting from basic linear expressions up to complex rational equations and domain restrictions.

Example 1: Linear Function Inverse

Question: Find the inverse of the function $f(x) = 4x - 7$ , and evaluate $f^{-1}(5)$ .

Step-by-step Solution:

Replace f(x) with y:

y = 4x - 7

Rearrange to make x the subject:

y + 7 = 4x \implies x = \dfrac{y + 7}{4}

Swap variables to find the inverse function:

f^{-1}(x) = \dfrac{x + 7}{4}

Evaluate at x = 5:

f^{-1}(5) = \dfrac{5 + 7}{4} = \dfrac{12}{4} = 3

Example 2: Rational Function Inverse (Standard A-Level Template)

Question: The function $g(x)$ is defined by:

g(x) = \dfrac{2x + 3}{x - 5}, \quad x \neq 5

Find an expression for $g^{-1}(x)$ and state the value of x for which $g^{-1}(x)$ is undefined.

Step-by-step Solution:

Replace $g(x)$ with $y$ :

y = \dfrac{2x + 3}{x - 5}

Multiply both sides by the denominator to clear the fraction:

y(x - 5) = 2x + 3 \implies xy - 5y = 2x + 3

Group all terms containing $x$ on one side and the rest on the other:

xy - 2x = 5y + 3

Factorise out $x$ to isolate it:

x(y - 2) = 5y + 3 \implies x = \dfrac{5y + 3}{y - 2}

Swap variables to write the inverse function:

g^{-1}(x) = \dfrac{5x + 3}{x - 2}

Identify where the inverse is undefined: An algebraic fraction is undefined when its denominator equals zero.

x - 2 = 0 \implies x = 2

Therefore, $g^{-1}(x)$ is undefined at $x = 2$ (the domain of the inverse is $x \neq 2$ , which is the range of the original function $g(x)$ ).

Examiner Tips & Common Pitfalls

Examiner Tip: When finding the inverse of a rational function (like in Example 2), students often make algebraic mistakes when factorising out x. Always group your x-terms on one side of the equals sign before attempting to factorise. If you have $xy - 2x = 5y + 3$ , you cannot divide by y or 2 individually; you must write $x(y - 2)$ first.

Additionally, always double-check if your calculated domain of the inverse function is mathematically valid for your expression. For instance, if your inverse contains a square root $\sqrt{x - a}$ , your domain must restrict $x \ge a$ to keep the root real.

Practice Questions & Solutions

Score:

The function f(x) is defined by:

$\text{[math]}$

for

$\text{[math]}$ .

Find

$\text{[math]}$

and state its domain.

Solution

Step 1:

$\text{[math]}$

Step 2: Multiply out:

$\text{[math]}$

Step 3: Rearrange to isolate x:

$\text{[math]}$

Step 4: Swap variables:

$\text{[math]}$

Step 5: Determine the domain. The denominator cannot be zero, so the domain of the inverse is:

$\text{[math]}$

(which is also the range of the original function f(x) because a fraction with a constant numerator 3 can never equal zero).

The function g(x) is defined as:

$\text{[math]}$

for

x greater than 3.

Find the expression for

$\text{[math]}$

and state its range.

Solution

Step 1:

$\text{[math]}$

Step 2: Rearrange to isolate x by converting the logarithmic form to exponential form:

$\text{[math]}$

Step 3: Swap variables:

$\text{[math]}$

Step 4: State the range. The range of the inverse function is identical to the domain of the original function. Since g(x) is defined for x > 3, the range of the inverse is: $\text{[math]}$

(which matches the output of $\text{[math]}$ because $\text{[math]}$ for all real values of x).

A function is defined as:

$\text{[math]}$

for

$\text{[math]}$ .

Find

$\text{[math]}$

by completing the square first.

Solution

Step 1: Complete the square on the original quadratic:

$\text{[math]}$

Step 2: Set equal to y:

$\text{[math]}$

Step 3: Rearrange to isolate x:

$\text{[math]}$

(Note: We select the positive root because the domain restriction is $\text{[math]}$ ).

Step 4: Swap variables:

$\text{[math]}$

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Gianpiero Placidi

UK-based Chemistry graduate with a passion for education, providing clear explanations and thoughtful guidance to inspire student success.

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Chapters

Core Principles of Inverse Functions
The One-to-One (Bijective) Requirement
Step-by-Step Method to Calculate the Inverse
Exam-Style Worked Examples
Examiner Tips & Common Pitfalls
Practice Questions & Solutions

What is the Inverse of a Function?

Core Principles of Inverse Functions

The Domain and Range Swap

The Identity Composition Rule

Graphical Symmetry

The One-to-One (Bijective) Requirement

Step-by-Step Method to Calculate the Inverse

Exam-Style Worked Examples

Example 1: Linear Function Inverse

Example 2: Rational Function Inverse (Standard A-Level Template)

Examiner Tips & Common Pitfalls

Practice Questions & Solutions

Theory

Exponent Rules

Composition of Functions

Constant Functions

Convergent and Divergent Sequences

Even and Odd Functions

Exponential Function

Graphing Parabolas

Arithmetic Sequence

Mean Value Theorem

Rolle’s Theorem

Intervals of Increase and Decrease

Periodic Functions

Graphing rational function I

Graphing rational function III

Inflection Points

Increasing and Decreasing Functions

Piecewise Functions

Trigonometric Functions

Linear Function

Domain of a Function

Maxima, Minima and Inflection Points

Linear Functions Worksheet

Monotone Sequence

Radical Functions

Function

Geometric Sequence

Coordinates of X Intercept Explained: GCSE and A Level

Bounded Sequence: Monotonic and Non-Monotic

L’Hôpital’s Rule

Graphing polynomial function I

Types of Functions

Coordinates in the Plane

Optimization

nth Term

Graphing rational function V

Limit of a Sequence

Absolute Value Function

Sequences

Geometric Sequence Problems And Solutions

Arithmetic Sequence Problems with Solutions

Logarithmic Functions

Concave and Convex Functions: Theory and Examples

Rational Functions: Graphs, Hyperbolas and Asymptotes Explained

Bounded Function Examples: Theory and Practice Questions

Maxima and Minima Examples With Solutions

Asymptote Functions: Theory and Practice Problems

Absolute Maxima Explained: Master Max/Min On Intervals

Inverse Function Explained: Simple Steps for A-Level Maths

Graphs of Quadratic Equations: Practice Questions and Answers

Applying Limits in Math: Limit Laws Made Clear

Formulas

Sequence Formulas | Arithmetic, Geometric and More

Exercises

Inflection Points Worksheet

Sequences Problems

Concavity and Convexity Worksheet

Increase and Decrease Worksheet

Limit of Sequence Problems

Applications of Derivatives Worksheet

Inverse Functions Worksheet

Exponential, Logarithmic and Trigonometric Functions Worksheet

Composition of Functions Worksheet

Graphing Worksheet

Mean Value Theorem Word Problems

Maximum and Minimum Word Problems

Quadratic Function Word Problems

Rational Functions Worksheet