Inverse relations and functions

Overview

An inverse relation reverses a relation by swapping each input and output. If a function sends $2$ to $7$, its inverse relation sends $7$ back to $2$.

This topic matters because inverse thinking is used to undo operations, solve equations, interpret domain and range, and connect algebraic rules with graphs. Learners also need to know that the inverse of a function is not always a function.

The big idea is "undoing in the opposite direction." A relation tells us where an input goes. An inverse relation asks where that output came from. This is why inverse work must keep input-output order very clear.

Prerequisites

• Relations and functions - Inverses reverse the input-output connections of relations and functions.
• Domain and range - The domain and range swap when an inverse relation is formed.
• Algebraic expressions and equations - Finding inverse rules requires rearranging equations.
• Linear simultaneous equations - Equation-solving confidence helps when isolating variables.
• Graphs of relations and functions - Inverse graphs are related by reflection in the line $y=x$.

Learning Scope

This chapter covers inverse relations from ordered pairs, inverse functions from rules, the notation $f^{-1}$, checking inverse results by substitution or composition, domain-range swapping, and deciding whether the inverse of a function is also a function.

This page does not teach inverse matrices; that is a separate Form IV matrix topic. It also does not fully teach graph reflection, which belongs mainly to Graphs of relations and functions.

Subtopics

Inverse Relations From Ordered Pairs

To form an inverse relation, swap each ordered pair:

$$ (x,y) \rightarrow (y,x) $$

If:

$$ R=\{(1,4),(2,6),(3,8)\} $$

then:

$$ R^{-1}=\{(4,1),(6,2),(8,3)\} $$

Key insight: The inverse relation reverses direction. Inputs become outputs and outputs become inputs.

Why this works: the original pair $(x,y)$ says "start at $x$ and arrive at $y$." The inverse pair $(y,x)$ says "start at $y$ and return to $x$." Nothing new is being calculated; the direction of the connection is being reversed.

Checking routine:

1. Swap the two entries in every ordered pair.
2. Count the number of pairs to make sure none were lost.
3. Check whether the old range is now the new domain.

Inverse Functions

An inverse function is an inverse relation that is also a function. This means each input in the inverse has exactly one output.

If:

$$ f=\{(1,3),(2,5),(3,7)\} $$

then:

$$ f^{-1}=\{(3,1),(5,2),(7,3)\} $$

This inverse is a function because no input in $f^{-1}$ is repeated with different outputs.

To test the inverse, look at the original outputs. If every original output is used only once, then after swapping, no input in the inverse will be forced to choose between two outputs.

When An Inverse Is Not A Function

Consider:

$$ g=\{(1,4),(2,4),(3,5)\} $$

This is a function because each input has exactly one output. Its inverse relation is:

$$ g^{-1}=\{(4,1),(4,2),(5,3)\} $$

Now the input $4$ has two different outputs, $1$ and $2$. Therefore $g^{-1}$ is not a function.

Key insight: A function must be one-to-one for its inverse relation to be a function.

One-to-one means different inputs give different outputs. If two inputs share an output in the original function, that shared output becomes a repeated input in the inverse. That is the warning sign that the inverse relation will not be a function.

Finding An Inverse Rule

To find the inverse of a rule:

1. Write $y=f(x)$.
2. Swap $x$ and $y$.
3. Solve for $y$.
4. Write the result as $f^{-1}(x)$.

For a linear function:

$$ f(x)=2x+3 $$

write:

$$ y=2x+3 $$

Swap:

$$ x=2y+3 $$

Solve:

$$ \begin{aligned} x-3 &= 2y \\ y &= \frac{x-3}{2} \end{aligned} $$

So:

$$ f^{-1}(x)=\frac{x-3}{2} $$

Why the swap step is necessary: the original rule starts with $x$ and produces $y$. The inverse must start with the old $y$ and recover the old $x$. Swapping $x$ and $y$ rewrites the equation from that reversed point of view.

After solving, the final variable is usually renamed as $x$ in $f^{-1}(x)$. This does not undo the swap; it simply uses the standard input letter for the inverse rule.

Checking Inverses

If two functions are inverses, applying one after the other returns the starting value.

For $f(x)=2x+3$ and $f^{-1}(x)=\frac{x-3}{2}$:

$$ \begin{aligned} f^{-1}(f(x)) &= \frac{(2x+3)-3}{2} \\ &= \frac{2x}{2} \\ &= x \end{aligned} $$

This confirms the inverse rule.

It is also useful to check the other direction:

$$ \begin{aligned} f(f^{-1}(x)) &= 2\left(\frac{x-3}{2}\right)+3 \\ &= x-3+3 \\ &= x \end{aligned} $$

For two functions to be inverses, both directions return the starting input on the allowed domains.

Domain And Range Of An Inverse

The inverse swaps domain and range:

$$ \text{Domain of } f^{-1} = \text{Range of } f $$

and:

$$ \text{Range of } f^{-1} = \text{Domain of } f $$

This is why inverse questions often appear together with domain and range questions.

If the original function has a restricted domain, the inverse may have a restricted range. If the original function has a restricted range, the inverse may have a restricted domain. This is especially important for fractions and for functions whose inverse is only a function after a domain restriction.

Key Terms

• Inverse relation: A relation formed by swapping each input-output pair.
• Inverse function: An inverse relation that is also a function.
• $f^{-1}$: Notation for the inverse function or inverse relation of $f$.
• One-to-one function: A function in which different inputs have different outputs.
• Composition: Applying one function after another.
• Reflection line: For graphs of inverses, the line $y=x$ is the mirror line.

Worked Examples

Example 1: Find An Inverse Relation

Given:

$$ R=\{(-1,2),(0,4),(3,7)\} $$

Find $R^{-1}$.

Swap each ordered pair:

$$ (-1,2)\rightarrow(2,-1) $$

$$ (0,4)\rightarrow(4,0) $$

$$ (3,7)\rightarrow(7,3) $$

Therefore:

$$ R^{-1}=\{(2,-1),(4,0),(7,3)\} $$

Check:

• The original domain was $\{-1,0,3\}$.
• The original range was $\{2,4,7\}$.
• The inverse domain is $\{2,4,7\}$, so the swap is consistent.

Example 2: Find An Inverse Function

Find the inverse of:

$$ f(x)=5x-4 $$

Let $y=5x-4$ and swap $x$ and $y$:

$$ x=5y-4 $$

Solve for $y$:

$$ \begin{aligned} x+4 &= 5y \\ y &= \frac{x+4}{5} \end{aligned} $$

So:

$$ f^{-1}(x)=\frac{x+4}{5} $$

Check by composition:

$$ \begin{aligned} f^{-1}(f(x)) &= \frac{(5x-4)+4}{5} \\ &= \frac{5x}{5} \\ &= x \end{aligned} $$

So the inverse rule correctly undoes $f(x)=5x-4$.

Example 3: Evaluate An Inverse At A Number

Given:

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

Find $f^{-1}\left(\frac{1}{3}\right)$.

Let:

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

Swap:

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

Solve for $y$:

$$ \begin{aligned} x(y-2) &= 1 \\ xy-2x &= 1 \\ xy &= 1+2x \\ y &= \frac{1+2x}{x} \end{aligned} $$

So:

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

Now substitute $x=\frac{1}{3}$:

$$ \begin{aligned} f^{-1}\left(\frac{1}{3}\right) &= \frac{1+2\left(\frac{1}{3}\right)}{\frac{1}{3}} \\ &= \frac{\frac{5}{3}}{\frac{1}{3}} \\ &=5 \end{aligned} $$

Therefore:

$$ f^{-1}\left(\frac{1}{3}\right)=5 $$

Check using the original function:

$$ f(5)=\frac{1}{5-2}=\frac{1}{3} $$

Since $f(5)=\frac{1}{3}$, the inverse statement $f^{-1}\left(\frac{1}{3}\right)=5$ is correct.

Example 4: Decide Whether An Inverse Is A Function

Given:

$$ P=\{(1,2),(2,4),(3,4),(4,6)\} $$

First, check that $P$ is a function. The inputs are $1,2,3,4$, and each appears once, so $P$ is a function.

Now form the inverse:

$$ P^{-1}=\{(2,1),(4,2),(4,3),(6,4)\} $$

In $P^{-1}$, the input $4$ appears twice with different outputs:

$$ (4,2)\quad \text{and}\quad (4,3) $$

Therefore $P^{-1}$ is not a function.

Why this happened:

The original function had two different inputs, $2$ and $3$, with the same output $4$. After swapping, the inverse input $4$ had to point back to both $2$ and $3$.

Example 5: Find And Check An Inverse Of A Fractional Linear Rule

Find the inverse of:

$$ g(x)=\frac{x-1}{4} $$

Write $y=\frac{x-1}{4}$, then swap:

$$ x=\frac{y-1}{4} $$

Solve for $y$:

$$ \begin{aligned} 4x &= y-1 \\ 4x+1 &= y \end{aligned} $$

So:

$$ g^{-1}(x)=4x+1 $$

Check:

$$ \begin{aligned} g(g^{-1}(x)) &= \frac{(4x+1)-1}{4} \\ &= \frac{4x}{4} \\ &= x \end{aligned} $$

The inverse is correct.

Example 6: Use An Inverse To Find An Input

If $h(x)=2x-7$, find $h^{-1}(9)$.

The expression $h^{-1}(9)$ asks: "What input gives output $9$ under $h$?"

Set the original function equal to $9$:

$$ 2x-7=9 $$

Solve:

$$ \begin{aligned} 2x &= 16 \\ x &= 8 \end{aligned} $$

Therefore:

$$ h^{-1}(9)=8 $$

Check:

$$ h(8)=2(8)-7=16-7=9 $$

Common Mistakes

• Thinking $f^{-1}(x)$ means $\frac{1}{f(x)}$. Correction: $f^{-1}$ means inverse function, not reciprocal, unless the context specifically makes them the same. Warning sign: you are about to put the whole function in the denominator.
• Forgetting to swap $x$ and $y$ before solving. Correction: the inverse reverses input and output. Warning sign: your "inverse" still performs the original operations in the original order.
• Swapping only one ordered pair or losing a pair. Correction: swap every pair and count them. Warning sign: the inverse relation has fewer pairs than the original relation.
• Assuming every inverse relation is a function. Correction: check whether the inverse has repeated inputs with different outputs. Warning sign: the original function has repeated output values.
• Losing domain restrictions. Correction: remember that inverse domain comes from the original range. Warning sign: the original function has a fraction, square, square root, or stated interval restriction.
• Checking only with one number and treating that as proof. Correction: a numerical check is useful, but composition gives the stronger algebraic check.
• Confusing inverse functions with inverse matrices. Correction: the word inverse appears in more than one topic, but the objects are different. Warning sign: the object is a matrix rather than a function rule or relation.

Practice Tasks

Foundation

1. Find the inverse relation of $\{(1,2),(3,4),(5,6)\}$.
2. State the domain and range of the relation in Question 1, then state the domain and range of its inverse.
3. Explain in one sentence why $f^{-1}(x)$ is not the same notation as $\frac{1}{f(x)}$.

Skill-Building

1. Decide whether the inverse of $\{(1,5),(2,5),(3,7)\}$ is a function.
2. Find the inverse of $f(x)=3x+2$.
3. Find the inverse of $g(x)=\frac{x-1}{4}$.
4. If $h(x)=2x-7$, find $h^{-1}(9)$ and check using $h(x)$.

Exam-Style

1. Show that $f(x)=x+6$ and $g(x)=x-6$ are inverses by composition.
2. Find the inverse of $p(x)=\frac{2x+5}{3}$, then check one direction by composition.
3. Given $R=\{(-2,1),(0,3),(4,3),(5,8)\}$, find $R^{-1}$ and decide whether $R^{-1}$ is a function.

Challenge

1. Explain why a function with repeated output values may have an inverse relation that is not a function.
2. Create a function with four ordered pairs whose inverse is also a function. Explain the one-to-one evidence.
3. Create a function with four ordered pairs whose inverse is not a function. Explain the repeated-output evidence before and after swapping.

Generated Question Layer

• Ordered-pair questions: swap pairs and classify the inverse.
• Algebraic-rule questions: find inverse rules by rearranging equations.
• Evaluation questions: compute values such as $f^{-1}(a)$.
• Domain-range questions: state how the domain and range change under inversion.
• Reasoning questions: explain why one-to-one behavior matters.

Learner Aid Opportunities

• diagram: Build side-by-side mapping diagrams that show each arrow reversing and then mark whether the inverse passes the function test.
• graph: Prepare a graph-reflection support for $y=x$ as a future visual aid, with labels for swapped coordinates.
• interactive: Design an ordered-pair swap activity where learners must also classify the inverse as function or not function.
• LLM tutor: Use a guided routine that asks learners to name the original domain/range, swap pairs or variables, check repeated inverse inputs, and distinguish inverse notation from reciprocal notation.

Exam-Derived Signals

These signals are assessment leads, not verified official past-question links. They should be checked against original papers and marking schemes before being used as final learner-facing references.

| Source | Current Signal | Review Status | Use Carefully As | | --- | --- | --- | --- | | data/exam_format_topic_crosswalk_2022.jsonl | Official 2022 format group Linear Programming/Functions/Relations maps to this topic and sibling pages; one item; weight 7.14. | Official format mapping; topic-page use still unreviewed. | Evidence that inverse work sits inside the broader functions/relations format group. | | data/topic_frequency_2021_2025.json | topic-inverse-relations-and-functions is listed under low-or-no coverage topics in the mapped extraction summary. | Unreviewed aggregate. | A signal that direct extracted coverage may be sparse. | | data/question_map_2021_2025.jsonl | Secondary links include inverse wording in variation problems and a matrix-inverse item. | mapped_unreviewed; likely noisy. | Mapping leads only, not evidence of direct testing. | | data/question_map_2021_2025.jsonl | A 2022 item involving $f^{-1}\left(\frac{1}{3}\right)$ is mapped primarily to topic-relations-and-functions. | Needs manual review. | Candidate for future cross-linking if the original paper confirms inverse-function focus. |

Source And Review Notes

• Topic registry status: official in data/curriculum_map.json.
• Learner expansion status: original prose drafted from the official syllabus topic and local assessment signals.
• Exam mapping status: unreviewed except for the official exam-format crosswalk group.
• Review risk: extracted inverse signals may confuse inverse functions, inverse variation, and inverse matrices, so reviewed links should be assigned carefully.