A1.5 Determine numeric and graphical representations of inverse relations

Ontario Grade 11 Mathematics

Characteristics of Functions

MCR3U – A1.5 | Switching Inputs and Outputs

Every relation pairs inputs with outputs. But what happens when you flip that around — when yesterday's output becomes today's input? That swap creates a brand-new relation called the inverse relation. Inverse relations appear throughout Grade 11 mathematics: they connect a function to its mirror image, and they are the foundation for understanding inverse functions later in the course. This lesson focuses on two concrete ways to represent an inverse relation — using a table of values (the numeric way) and using a graph (the graphical way). No algebraic formula is needed yet; the goal is to build a clear picture of what "inverse" really means before any symbols get in the way.

What you will learn

Explain what an inverse relation is in plain language and using ordered pairs.
Switch the x- and y-values in a table of values to produce the inverse relation numerically.
Reflect a graph over the line y = x to produce the graphical representation of the inverse relation.
Determine whether the inverse relation is a function by applying the vertical-line test.

Prerequisite Bridge: Relations, Functions, and the Vertical-Line Test

A relation is any set of ordered pairs

(x, y)

. The first value,

x

, is the input, and the second value,

y

, is the output. A function is a special relation where every input has exactly one output — no

x

-value is repeated with a different

y

-value.

The vertical-line test is a quick graphical check: if every vertical line you can draw touches a graph at most once, the relation is a function. If any vertical line hits the graph at two or more points, the relation is not a function. You will use this test again at the end of the lesson to check whether an inverse relation is also a function.

One more tool you need: the line

y = x

. This is the diagonal line that passes through the origin at a 45-degree angle, hitting points like

(0, 0)

,

(1, 1)

,

(2, 2)

, and

(-3, -3)

. It acts as a mirror for graphing inverse relations.

A relation pairs inputs ( $x$ ) with outputs ( $y$ ).
A function has exactly one output for every input.
The vertical-line test checks whether a graph represents a function.
The line $y = x$ serves as the reflection mirror for inverse relations.

What Is an Inverse Relation?

The inverse relation of a given relation is formed by swapping every input and output. In other words, if the original relation contains the ordered pair

(a, b)

, then the inverse relation contains

(b, a)

. Every

x

-value becomes a

y

-value, and every

y

-value becomes an

x

-value.

Think of a practical situation: suppose a relation matches a student's study time (hours) to their quiz score. The inverse relation matches quiz scores back to study times. The information is identical — only the direction of the pairing has changed.

The notation used for the inverse of a relation

f

is

f^{-1}

(read as 'f inverse'). Be careful: the superscript

-1

here does not mean 'one divided by

f

'. It is a label that specifically means the inverse relation. At this stage in the course,

f^{-1}

represents the inverse relation, which may or may not itself be a function.

The clearest way to see an inverse relation is to start with a table of values, swap the two columns, and then look at what changed on a graph.

(a,\ b) \longrightarrow (b,\ a)

Swapping every $(x, y)$ to $(y, x)$ produces the inverse relation.
The inverse relation is labelled $f^{-1}$ ; the $-1$ is not an exponent.
The inverse relation may or may not be a function.
Both the numeric and graphical representations reveal the inverse clearly.

Numeric Representation: Swapping the Table of Values

A table of values lists ordered pairs in two columns. To find the inverse relation numerically, you simply exchange the role of the two columns: the

x

-column becomes the

y

-column, and the

y

-column becomes the

x

-column.

After swapping, re-read the table left-to-right as usual. Each row now gives you a new ordered pair that belongs to the inverse relation. Nothing about the numbers themselves changes — only their positions in the pair.

Once you have the swapped table, scan the new

x

-column. If any value appears more than once, the inverse relation is not a function (because one input would map to multiple outputs). If every value in the new

x

-column is unique, the inverse relation is a function.

This column-swap method is completely reliable and works for any relation, whether or not it is a function to begin with.

Swap the $x$ - and $y$ -columns to get the inverse relation's table.
Each row in the new table is one ordered pair of the inverse relation.
Check the new $x$ -column for repeated values to test whether the inverse is a function.
The column-swap method works for any relation.

Graphical Representation: Reflecting Over y = x

Swapping

x

and

y

in every ordered pair has a precise geometric meaning: it reflects each point across the line

y = x

. This is because the line

y = x

is equidistant from any point

(a, b)

and its mirror image

(b, a)

.

To draw the inverse relation's graph, you can use one of two approaches. First, you can plot the swapped ordered pairs directly from your table. Second, you can draw the original graph and then sketch its reflection across the line

y = x

, flipping the entire curve over that diagonal mirror line.

A useful check: the original graph and its inverse are symmetric about the line

y = x

. If you fold your graph paper along

y = x

, the two curves should line up exactly.

After drawing the inverse graph, apply the vertical-line test to it. If the original relation already passed the vertical-line test (i.e., it was a function), its inverse might not pass — it depends on whether the original graph also passes the horizontal-line test. If every horizontal line touches the original graph at most once, then the inverse relation will be a function.

y = x

Reflecting a graph over the line $y = x$ produces the graph of the inverse relation.
The original graph and its inverse are symmetric about $y = x$ .
Plot swapped pairs or flip the curve — both methods give the same result.
Apply the vertical-line test to the inverse graph to check if it is a function.

Is the Inverse Relation a Function? The Horizontal-Line Test

You already know the vertical-line test: a graph is a function if no vertical line crosses it more than once. There is a matching test for inverses called the horizontal-line test: if every horizontal line touches the original graph at most once, then the inverse relation will be a function.

Why does this work? A horizontal line on the original graph acts like a vertical line on the inverse graph (because the axes have been swapped). So if a horizontal line hits the original at two points, say

(2, 5)

and

(7, 5)

, the inverse will contain both

(5, 2)

and

(5, 7)

— two different outputs for the same input

x = 5

, which breaks the function rule.

To summarise the decision process: start with the original graph, apply the horizontal-line test, and use the result to predict whether the inverse is a function. Then confirm by applying the vertical-line test to the inverse graph itself.

The horizontal-line test applied to the original graph predicts whether its inverse is a function.
If any horizontal line crosses the original graph twice, the inverse relation is not a function.
Confirm by applying the vertical-line test directly to the inverse graph.

Side-by-Side: Original Relation g and Its Inverse g⁻¹

Original pair in g	Inverse pair in g⁻¹	What changed?
$(-2,\ 1)$	$(1,\ -2)$	x and y swapped
$(0,\ 1)$	$(1,\ 0)$	x and y swapped
$(1,\ 3)$	$(3,\ 1)$	x and y swapped
$(3,\ 5)$	$(5,\ 3)$	x and y swapped

Worked example

Example 1 – Numeric Inverse of a Simple Relation

A relation

f

is given by the following ordered pairs:

(1, 4)

,

(2, 7)

,

(3, 10)

,

(4, 13)

. (a) Write the ordered pairs of the inverse relation

f^{-1}

. (b) State whether

f^{-1}

is a function, and explain why.

List the original ordered pairs
Write out every pair from the relation $f$ so you can see both coordinates clearly before swapping.
$f: (1,4),\ (2,7),\ (3,10),\ (4,13)$
Swap each pair
For each ordered pair $(x, y)$ in $f$ , write $(y, x)$ in $f^{-1}$ . The first coordinate and second coordinate trade places.
$f^{-1}: (4,1),\ (7,2),\ (10,3),\ (13,4)$
Check the new x-values for repeats
Look at the first coordinate of every pair in $f^{-1}$ . The values are 4, 7, 10, and 13. All four are different, so no input maps to more than one output.
State the conclusion
Because every input in $f^{-1}$ is unique, the inverse relation passes the definition of a function.

Answer:

f^{-1} = \{(4,1),\ (7,2),\ (10,3),\ (13,4)\}

. The inverse relation is a function because no

x

-value repeats.

Check: To verify, swap the pairs of

f^{-1}

back: the result is

(1,4), (2,7), (3,10), (4,13)

, which matches

f

exactly. Also confirm that the

x

-values in

f^{-1}

— namely 4, 7, 10, and 13 — are all distinct. Both checks pass.

Worked example

Example 2 – Graphical Inverse and Testing Whether It Is a Function

A relation

g

has the table of values below. (a) Produce the table of values for

g^{-1}

. (b) Describe how to sketch the graph of

g^{-1}

from the graph of

g

. (c) Determine whether

g^{-1}

is a function.

|

x

|

y

|
|---|---|
|

-2

|

1

|
|

0

|

1

|
|

1

|

3

|
|

3

|

5

|

Write the original ordered pairs
Read each row of the table as an ordered pair.
$g: (-2,1),\ (0,1),\ (1,3),\ (3,5)$
Swap each pair to get the inverse table
Exchange the $x$ - and $y$ -values in every pair. The new table has columns labelled $x$ and $y$ , where the $x$ -values come from $g$ 's original $y$ -values.
$g^{-1}: (1,-2),\ (1,0),\ (3,1),\ (5,3)$
Describe the graphical method
Plot the four points of $g$ on a coordinate grid and draw the line $y = x$ as a dashed mirror. Then reflect each point across that line: a point $(a, b)$ maps to $(b, a)$ . Connecting the reflected points gives the graph of $g^{-1}$ .
Apply the vertical-line test to the inverse graph
Notice that the inverse relation contains $(1, -2)$ and $(1, 0)$ . Both have the input $x = 1$ but different outputs $y = -2$ and $y = 0$ . A vertical line at $x = 1$ would hit the graph of $g^{-1}$ at two points, so the vertical-line test fails.
State the conclusion
Because the input $x = 1$ produces two different outputs, the inverse relation $g^{-1}$ fails the definition of a function. This happened because $g$ itself had two different inputs ( $-2$ and $0$ ) that both produced the same output ( $1$ ), meaning $g$ failed the horizontal-line test.

Answer:

g^{-1} = \{(1,-2),\ (1,0),\ (3,1),\ (5,3)\}

. The graph of

g^{-1}

is the reflection of

g

over the line

y = x

. The inverse relation is NOT a function because the input

x = 1

maps to two different outputs.

Check: Swapping the pairs of

g^{-1}

back produces

(-2,1),(0,1),(1,3),(3,5)

, which matches

g

exactly. The repeated

x = 1

in

g^{-1}

confirms the vertical-line test fails. The original table of

g

shows

y = 1

appears twice (for

x = -2

and

x = 0

), confirming the horizontal-line test on

g

also fails. All checks are consistent.

Common mistakes and how to avoid them

Writing

f^{-1}

as

\frac{1}{f}

because the

-1

looks like an exponent.

Correction: In the context of inverse relations, the notation

f^{-1}

means the inverse relation, not one divided by

f

. It is a label, not a power.

Swapping only the x-values without touching the y-values, for example changing

(3, 7)

to

(-3, 7)

.

Correction: Forming the inverse requires swapping both coordinates completely:

(3, 7)

becomes

(7, 3)

. The original

y

-value becomes the new

x

-value.

Assuming the inverse of a function is always a function.

Correction: The inverse relation is a function only if the original graph passes the horizontal-line test. If any output in the original is repeated for different inputs, the inverse will not be a function.

Reflecting the graph over the x-axis or y-axis instead of over the line

y = x

.

Correction: The correct mirror line is the diagonal

y = x

. Reflecting over the x-axis gives

(x, -y)

and over the y-axis gives

(-x, y)

— neither produces the inverse relation.

Forgetting to re-label the axes after swapping, then misreading which variable is the input.

Correction: After swapping, the new left column is always the

x

(input) column and the new right column is always the

y

(output) column. Re-labelling prevents confusion when applying the vertical-line test.

Lesson summary

The inverse relation is formed by swapping every ordered pair $(x, y)$ to $(y, x)$ , exchanging all inputs and outputs.
Numerically, swap the two columns in a table of values; each new row is one ordered pair of the inverse relation.
Graphically, reflect the original graph across the line $y = x$ ; the image is the graph of the inverse relation.
The notation $f^{-1}$ labels the inverse relation — the superscript $-1$ is not an exponent and does not mean $\frac{1}{f}$ .
Apply the vertical-line test to the inverse graph to decide whether the inverse relation is also a function.
The horizontal-line test on the original graph predicts the same result: if it passes, the inverse will be a function.

Check your understanding

Question 1

A relation contains the ordered pair

(5, -3)

. Which ordered pair must appear in the inverse relation?

$(-5, 3)$
$(-3, 5)$
$(5, 3)$
$(-5, -3)$

Show answer and explanation

(-3, 5)

Forming the inverse swaps the two coordinates:

(5, -3)

becomes

(-3, 5)

. The signs of the numbers do not change — only the positions of

x

and

y

are exchanged.

Question 2

The graph of a relation passes the vertical-line test but fails the horizontal-line test. What can you conclude about its inverse relation?

The inverse relation is a function.
The inverse relation is not a relation.
The inverse relation is not a function.
The inverse relation is the same as the original.

Show answer and explanation

The inverse relation is not a function.

Failing the horizontal-line test means some output value is produced by more than one input. After swapping, that repeated output becomes a repeated input in the inverse, causing the inverse to fail the vertical-line test. Therefore the inverse relation is not a function.

Question 3

Which line acts as the mirror when reflecting a graph to produce its inverse relation?

$y = 0$
$x = 0$
$y = -x$
$y = x$

Show answer and explanation

y = x

Swapping

x

and

y

in every ordered pair is geometrically equivalent to reflecting each point across the line

y = x

. That diagonal line is equidistant from any point

(a, b)

and its swapped image

(b, a)

.

Question 4

A table of values for

f^{-1}

has the following new

x

-values after swapping: 2, 5, 5, 9. Is

f^{-1}

a function?

Yes, because all original $x$ -values were different.
Yes, because the $y$ -values are all different.
No, because the $x$ -value $5$ appears twice.
No, because the table has four rows.

Show answer and explanation

No, because the

x

-value

5

appears twice.

A function requires every input to have exactly one output. Here,

x = 5

appears twice in the new

x

-column, meaning the input

5

maps to two different outputs. That violates the definition of a function, so

f^{-1}

is not a function.

Key terms

Relation: A set of ordered pairs (x, y) that pairs input values with output values. There are no restrictions on how many outputs one input may have.
Function: A relation in which every input (x-value) is paired with exactly one output (y-value). No x-value may repeat with a different y-value.
Inverse relation: The relation formed by swapping the x- and y-coordinates of every ordered pair in the original relation, exchanging all inputs and outputs.
Ordered pair: A pair of numbers written as (x, y), where x is the input (horizontal position) and y is the output (vertical position).
Vertical-line test: A visual test applied to a graph: if every vertical line touches the graph at most once, the relation is a function.
Horizontal-line test: A visual test applied to the original graph: if every horizontal line touches the graph at most once, the inverse relation will be a function.
Reflection over y = x: A transformation that maps each point (a, b) to the point (b, a), using the line y = x as the mirror. This is exactly the graphical effect of forming an inverse relation.
f⁻¹ (f inverse): The notation for the inverse relation of f. The superscript −1 is a label meaning 'inverse', not an exponent; it does not mean one divided by f.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation A1.5. It is a study resource, not an official curriculum publication.

Official curriculum reference

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A1.5 · Determine numeric and graphical representations of inverse relations

Ontario Grade 11 Mathematics

MCR3U – A1.5 | Switching Inputs and Outputs

What you will learn

Prerequisite Bridge: Relations, Functions, and the Vertical-Line Test

What Is an Inverse Relation?

Numeric Representation: Swapping the Table of Values

Graphical Representation: Reflecting Over y = x

Is the Inverse Relation a Function? The Horizontal-Line Test

Side-by-Side: Original Relation g and Its Inverse g⁻¹

Example 1 – Numeric Inverse of a Simple Relation

Example 2 – Graphical Inverse and Testing Whether It Is a Function

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MCR3U

About this lesson