Inverse Functions

Understanding Inverse Functions

Every time you tie your shoelaces, you can untie them by performing the exact reverse sequence of movements. In mathematics, an inverse function performs the reverse process to "undo" the operations of the original function. If a regular function maps an input $a$ to an output $b$ so that $f(a) = b$ , the inverse function maps $b$ back to $a$ so that $f^{-1}(b) = a$ .

The standard formal notation for an inverse function is $f^{-1}(x)$ . It can also be expressed using mapping notation as $f^{-1}: x \mapsto \dots$ . Applying a function and then immediately applying its inverse returns you to your original starting value, which gives us the identity property $f(f^{-1}(x)) = x$ .

For simple linear functions, you can think of a function machine where you "reverse the arrows" and use opposite operations. For example, if $f(x) = 3x - 5$ , the forward process is $\times 3$ then $-5$ . The reverse process is $+5$ then $\div 3$ , giving an algebraic result of $f^{-1}(x) = \frac{x + 5}{3}$ .

Finding Inverse Functions Algebraically

While function machines work for simple equations, harder exam questions require a formal algebraic method. To find $f^{-1}(x)$ , the original function $f(x)$ must be rearranged to make $x$ the subject of a formula.

A highly recommended method to reduce algebraic errors is the "swap" method. First, replace $f(x)$ with $y$ , then swap the $x$ and $y$ variables entirely before rearranging the equation to isolate the new $y$ .

Find the inverse of $f(x) = 2x + 1$ .

Step 1: Write the function as an equation using $y$ .

$y = 2x + 1$

Step 2: Swap the $x$ and $y$ variables.

$x = 2y + 1$

Step 3: Rearrange to isolate $y$ .

$x - 1 = 2y$
$y = \frac{x - 1}{2}$

Step 4: State the final answer using inverse notation.

$f^{-1}(x) = \frac{x - 1}{2}$

Inverses with Algebraic Fractions

Questions where the variable appears twice, such as in algebraic fractions, are common at Higher Tier. These require expanding brackets and factorising to successfully change the subject.

Find $f^{-1}(x)$ for $f(x) = \frac{x+5}{x+3}$ .

Step 1: Set the expression to $y$ and swap the variables.

$x = \frac{y+5}{y+3}$

Step 2: Multiply by the denominator to clear the fraction.

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

Step 3: Expand the brackets.

$xy + 3x = y + 5$

Step 4: Collect all terms containing $y$ on one side of the equation.

$xy - y = 5 - 3x$

Step 5: Factorise out $y$ .

$y(x - 1) = 5 - 3x$

Step 6: Divide to isolate $y$ and write the final inverse function.

$y = \frac{5 - 3x}{x - 1}$
$f^{-1}(x) = \frac{5 - 3x}{x - 1}$

Domain and Range of Inverse Functions

Why do some functions lack an inverse unless we strictly limit the input numbers? For an inverse function to exist, the original function must be a one-to-one mapping, meaning each input maps to a unique output.

Many-to-one functions, such as $f(x) = x^2$ , require a restricted domain (e.g., $x \ge 0$ ) to become one-to-one so that a unique inverse can be defined. You must also exclude values that result in an undefined output, such as values that cause division by zero or negative square roots.

The most important rule connecting a function and its inverse is that their inputs and outputs are directly swapped. The domain of $f^{-1}(x)$ is exactly the range of $f(x)$ , and the range of $f^{-1}(x)$ is exactly the domain of $f(x)$ .

Graphs of Inverse Functions

When plotted on a coordinate grid, the graphical relationship between a function and its inverse is a visual reflection. The graph of $y = f^{-1}(x)$ is always a reflection of the graph $y = f(x)$ in the diagonal line $y = x$ .

Because the $x$ and $y$ values swap, any coordinate point $(a, b)$ that lies on the graph of $f(x)$ will transform into the point $(b, a)$ on the graph of $f^{-1}(x)$ . For example, if $f(x)$ crosses the $y$ -axis at $(0, 3)$ , $f^{-1}(x)$ must cross the $x$ -axis at $(3, 0)$ .

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Exam Tips

Common Mistake
Students frequently confuse the inverse notation $f^{-1}(x)$ with the reciprocal $\frac{1}{f(x)}$ — the ' $-1$ ' is just a label, not a negative power.
2
When asked to evaluate something like $f^{-1}(10)$ , you can often save time by solving the equation $f(x) = 10$ directly, rather than finding the full algebraic inverse first.
3
In Edexcel exams, you must strictly use the variable $x$ when stating the domain (e.g., $x \neq 3$ ) and use $f(x)$ or $y$ when stating the range (e.g., $f(x) > 0$ ) to avoid dropping marks.
4
If asked to sketch $y = f^{-1}(x)$ on an exam graph, always draw the diagonal line $y = x$ first to act as your mirror line for the reflection.
5
When rearranging to find the inverse of an algebraic fraction, watch out for sign errors when moving terms across the equals sign, as this is the most common cause of lost marks.

Key Terms(6)

Inverse function: A function that reverses the effect of another function, mapping an output value back to its original input value.
Reverse process: The sequence of opposite mathematical operations performed in the opposite order to return to a starting value.
Subject of a formula: The single variable, usually isolated on the left-hand side, that is expressed in terms of other variables.
Domain: The complete set of all possible input values ( $x$ -values) for which a function is mathematically defined.
Range: The complete set of all possible output values ( $y$ -values or $f(x)$ ) that a function can produce.
One-to-one mapping: A mathematical relationship where each input corresponds to exactly one unique output, and each output comes from exactly one unique input.

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Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(6)

Inverse function: A function that reverses the effect of another function, mapping an output value back to its original input value.
Reverse process: The sequence of opposite mathematical operations performed in the opposite order to return to a starting value.
Subject of a formula: The single variable, usually isolated on the left-hand side, that is expressed in terms of other variables.
Domain: The complete set of all possible input values ( $x$ -values) for which a function is mathematically defined.
Range: The complete set of all possible output values ( $y$ -values or $f(x)$ ) that a function can produce.
One-to-one mapping: A mathematical relationship where each input corresponds to exactly one unique output, and each output comes from exactly one unique input.

Inverse Functions

Understanding Inverse Functions

Finding Inverse Functions Algebraically

Find the inverse of $f(x) = 2x + 1$ .

Step 1: Write the function as an equation using $y$ .

$y = 2x + 1$

Step 2: Swap the $x$ and $y$ variables.

$x = 2y + 1$

Step 3: Rearrange to isolate $y$ .

$x - 1 = 2y$
$y = \frac{x - 1}{2}$

Step 4: State the final answer using inverse notation.

$f^{-1}(x) = \frac{x - 1}{2}$

Inverses with Algebraic Fractions

Questions where the variable appears twice, such as in algebraic fractions, are common at Higher Tier. These require expanding brackets and factorising to successfully change the subject.

Find $f^{-1}(x)$ for $f(x) = \frac{x+5}{x+3}$ .

Step 1: Set the expression to $y$ and swap the variables.

$x = \frac{y+5}{y+3}$

Step 2: Multiply by the denominator to clear the fraction.

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

Step 3: Expand the brackets.

$xy + 3x = y + 5$

Step 4: Collect all terms containing $y$ on one side of the equation.

$xy - y = 5 - 3x$

Step 5: Factorise out $y$ .

$y(x - 1) = 5 - 3x$

Step 6: Divide to isolate $y$ and write the final inverse function.

$y = \frac{5 - 3x}{x - 1}$
$f^{-1}(x) = \frac{5 - 3x}{x - 1}$

Domain and Range of Inverse Functions

Graphs of Inverse Functions

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students frequently confuse the inverse notation $f^{-1}(x)$ with the reciprocal $\frac{1}{f(x)}$ — the ' $-1$ ' is just a label, not a negative power.
2
When asked to evaluate something like $f^{-1}(10)$ , you can often save time by solving the equation $f(x) = 10$ directly, rather than finding the full algebraic inverse first.
3
In Edexcel exams, you must strictly use the variable $x$ when stating the domain (e.g., $x \neq 3$ ) and use $f(x)$ or $y$ when stating the range (e.g., $f(x) > 0$ ) to avoid dropping marks.
4
If asked to sketch $y = f^{-1}(x)$ on an exam graph, always draw the diagonal line $y = x$ first to act as your mirror line for the reflection.
5
When rearranging to find the inverse of an algebraic fraction, watch out for sign errors when moving terms across the equals sign, as this is the most common cause of lost marks.

Key Terms(6)

Inverse function: A function that reverses the effect of another function, mapping an output value back to its original input value.
Reverse process: The sequence of opposite mathematical operations performed in the opposite order to return to a starting value.
Subject of a formula: The single variable, usually isolated on the left-hand side, that is expressed in terms of other variables.
Domain: The complete set of all possible input values ( $x$ -values) for which a function is mathematically defined.
Range: The complete set of all possible output values ( $y$ -values or $f(x)$ ) that a function can produce.
One-to-one mapping: A mathematical relationship where each input corresponds to exactly one unique output, and each output comes from exactly one unique input.

Previous NoteIntroduction to Function Notation Next NoteComposite Functions

Back to Functions

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(6)

Inverse function: A function that reverses the effect of another function, mapping an output value back to its original input value.
Reverse process: The sequence of opposite mathematical operations performed in the opposite order to return to a starting value.
Subject of a formula: The single variable, usually isolated on the left-hand side, that is expressed in terms of other variables.
Domain: The complete set of all possible input values ( $x$ -values) for which a function is mathematically defined.
Range: The complete set of all possible output values ( $y$ -values or $f(x)$ ) that a function can produce.
One-to-one mapping: A mathematical relationship where each input corresponds to exactly one unique output, and each output comes from exactly one unique input.