A. Characteristics of Functions · Functions and Equivalent Expressions

Domain and Range: Inputs, Outputs and Real-World Limits

A1.3About 70 minutes

Learning goals

I can explain the meaning of domain and range.
I can state the domain and range from a list of points, a graph or an equation using set notation.
I can use open and closed dots and arrows correctly when reading a graph.
I can explain restrictions on the domain and range in real-world situations.

Words to know

domain
The set of all possible input values (x-values) of a relation.

range
The set of all possible output values (y-values) of a relation.

set notation
A way of describing a set, such as {x ∈ ℝ | x ≥ 2}: x is a real number such that x is at least 2.

ℝ
The set of real numbers.

continuous / discrete
A continuous graph is unbroken (any value in an interval is allowed). A discrete graph is a set of separate points.

restriction
A limit on the domain or range, from the mathematics (e.g. no division by zero) or the context (e.g. time ≥ 0).

Which inputs make sense?

A ski hill lift ticket machine accepts ages from 0 to 120 and prints a price. Typing an age of −4 or 3.7 million makes no sense. Every function comes with a set of inputs that are allowed and a set of outputs it can actually produce. These two sets have names: the domain and the range.

Definitions

The domain is the set of all possible input values (x-values).
The range is the set of all possible output values (y-values).

In this course we usually describe these sets with set notation:

{x ∈ ℝ} is read "x is an element of the real numbers": every real number is allowed.
{x ∈ ℝ | x ≥ −2} is read "x is an element of the real numbers such that x is greater than or equal to −2".
{1, 2, 3, 4} lists the values when there are only a few.

The symbol ℝ stands for the set of real numbers, which you met in Grade 9 when you built the real number system.

Words for the symbols

∈ means "is an element of" (belongs to). The vertical bar | means "such that". ℝ is all real numbers. ≤ and ≥ include the endpoint; < and > do not.

Domain and range from ordered pairs

If a function is a short list of points, just collect the x-values for the domain and the y-values for the range. Write each value once, usually in order from smallest to largest.

Worked example

Example 1: From a list

State the domain and range of f = {(−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4)}.

Domain: the x-values are −2, −1, 0, 1, 2. D = {−2, −1, 0, 1, 2}.
Range: the y-values are 4, 1, 0, 1, 4. Write each once: R = {0, 1, 4}.
Notice the range has fewer values than the domain. That happens with many-to-one functions.

Domain {−2, −1, 0, 1, 2}; range {0, 1, 4}.

Domain and range from a graph

Think of shining a light straight down on the graph: the shadow on the x-axis is the domain. Shine a light from the side: the shadow on the y-axis is the range.

Watch the ends of the graph:

A closed dot (●) means the endpoint is included: use ≤ or ≥.
An open dot (○) means the endpoint is not included: use < or >.
An arrowhead, or a curve running off the grid, means the graph keeps going forever in that direction.

The range is not always found at the endpoints. A parabola's vertex can give the lowest or highest y-value.

The graph of y = (x − 1)² − 4 for −2 ≤ x < 4. The left end is closed and the right end is open. The lowest point is the vertex, so the range starts at −4. The range still includes 5, because the closed dot at x = −2 reaches it.

Worked example

Example 2: Reading graphs

State the domain and range.

(a) The line y = 2x − 1 with arrows at both ends. (b) The parabola y = −x² + 3 with arrows at both ends. (c) A line segment from (−1, 4) (closed dot) to (5, 1) (open dot).

(a) A non-horizontal line with arrows extends forever left, right, up and down. Domain {x ∈ ℝ}, range {y ∈ ℝ}.
(b) The parabola opens down with vertex (0, 3). It extends forever left and right, but never goes above y = 3. Domain {x ∈ ℝ}, range {y ∈ ℝ | y ≤ 3}.
(c) x runs from −1 (included) to 5 (not included): domain {x ∈ ℝ | −1 ≤ x < 5}.
y goes from 4 (included, at x = −1) down towards 1 (not included, open dot): range {y ∈ ℝ | 1 < y ≤ 4}.

(a) D = {x ∈ ℝ}, R = {y ∈ ℝ}; (b) D = {x ∈ ℝ}, R = {y ∈ ℝ | y ≤ 3}; (c) D = {x ∈ ℝ | −1 ≤ x < 5}, R = {y ∈ ℝ | 1 < y ≤ 4}.

Domain and range from an equation

For polynomials such as linear and quadratic functions, any real number can go in, so the domain is {x ∈ ℝ}. The range takes more thought.

A non-horizontal linear function such as f(x) = 3x − 7 can produce every real number: range {y ∈ ℝ}.
A quadratic in vertex form y = a(x − h)² + k has its vertex at (h, k). If a > 0 it opens up, so the range is {y ∈ ℝ | y ≥ k}. If a < 0 it opens down: {y ∈ ℝ | y ≤ k}.

For example, y = x² + 1 has domain {x ∈ ℝ} and range {y ∈ ℝ | y ≥ 1}, because x² is never negative, so x² + 1 is never less than 1. Next lesson you will meet functions whose domain is restricted too.

Worked example

Example 3: Quadratics

State the domain and range of (a) f(x) = 2(x + 3)² − 5 (b) g(x) = −(x − 4)² + 7 (c) h(x) = x² − 6x + 10.

(a) Vertex (−3, −5), a = 2 > 0, opens up. D = {x ∈ ℝ}, R = {y ∈ ℝ | y ≥ −5}.
(b) Vertex (4, 7), a = −1 < 0, opens down. D = {x ∈ ℝ}, R = {y ∈ ℝ | y ≤ 7}.
(c) Complete the square (Grade 10): x² − 6x + 10 = (x² − 6x + 9) + 1 = (x − 3)² + 1. Vertex (3, 1), opens up. D = {x ∈ ℝ}, R = {y ∈ ℝ | y ≥ 1}.

All three have domain {x ∈ ℝ}; the ranges are y ≥ −5, y ≤ 7 and y ≥ 1.

Restrictions in real-world situations

In a real situation, the domain and range are often smaller than the mathematics alone allows. Ask:

Can the input be negative? (Time before the start, a negative number of tickets?)
Must the input be a whole number? (People, tickets, cars.) Then the graph is a set of separate points (discrete), not a smooth curve (continuous).
Is there a maximum? (Seats in an arena, the moment a ball hits the ground.)
What physical limits apply to the output? (A height cannot go below the ground.)

Worked example

Example 4: The thrown ball

A ball is thrown upward from a deck. Its height in metres after t seconds is h(t) = −5t² + 20t + 1. It reaches its greatest height, 21 m, at t = 2 and lands at about t = 4.05 s. What physical factors restrict the domain and range? State them.

Domain. Time starts when the ball is thrown, so t ≥ 0. The model stops being true when the ball hits the ground at t ≈ 4.05. So D = {t ∈ ℝ | 0 ≤ t ≤ 4.05}.
Without the context, the quadratic would accept any t, including negative times and times after landing, when the formula gives negative heights (underground!).
Range. The ball is never below the ground (h ≥ 0) and never higher than the maximum 21 m. So R = {h ∈ ℝ | 0 ≤ h ≤ 21}.
Other factors that could change the model: air resistance, wind, and the ball hitting a tree branch or the deck.

D = {t ∈ ℝ | 0 ≤ t ≤ 4.05}, R = {h ∈ ℝ | 0 ≤ h ≤ 21}.

Only the part of the parabola from the throw (t = 0) to the landing (t ≈ 4.05 s) describes the ball. That piece gives the domain and range in context.

Worked example

Example 5: A discrete situation

A school drama club sells tickets for $8 each. The gym holds 250 people. Revenue is R(n) = 8n, where n is the number of tickets sold. State the domain and range.

You cannot sell part of a ticket or a negative number of tickets, and no more than 250 fit. So n is a whole number from 0 to 250: D = {n ∈ 𝕎 | 0 ≤ n ≤ 250} (𝕎 is the set of whole numbers 0, 1, 2, ...).
The revenue goes up in steps of $8: 0, 8, 16, ..., 2000. So R = {0, 8, 16, ..., 2000}, the multiples of 8 from 0 to 2000.
The graph is discrete: 251 separate points, not a solid line.

D = {0, 1, 2, ..., 250}; R = {0, 8, 16, ..., 2000}.

Just for fun

A domain is a bit like a nightclub bouncer for numbers: "Sorry, −3, you're not on the list." The range is the list of who actually got out the other side.

Hands-on activity

Shadows of a graph

You need: grid paper, a ruler, a flashlight (or phone light) and a pencil.

Draw axes and graph y = (x − 1)² − 4 for −2 ≤ x < 4 using the points x = −2, −1, 0, 1, 2, 3 and an open dot at (4, 5).
Lay a ruler along the x-axis. Shine the light from straight above and imagine each point dropping down to the x-axis. Shade the x-axis from −2 to 4 (closed dot at −2, open dot at 4).
Now imagine each point sliding sideways to the y-axis. Shade the y-axis from the lowest to the highest point reached.
Write the domain and range in set notation.

What to look for: points (−2, 5), (−1, 0), (0, −3), (1, −4), (2, −3), (3, 0), and an open dot at (4, 5). Domain {x ∈ ℝ | −2 ≤ x < 4}; range {y ∈ ℝ | −4 ≤ y ≤ 5}. The range includes 5 because (−2, 5) is a closed dot, even though (4, 5) is open.

Your turn

State the domain and range of {(3, 7), (4, 7), (5, 8), (6, 9)}.

Read aloud: {x ∈ ℝ | x < 6}.

State the domain and range of f(x) = 5x + 2.

State the domain and range of g(x) = (x − 2)² + 3.

State the domain and range of h(x) = −4x² + 1.

A segment runs from (−3, −1) (open dot) to (2, 9) (closed dot). State its domain and range.

What is the range of the constant function y = −4?

More practice: warm-up

Is the domain made of inputs or outputs?

What does an open dot at the end of a graph mean?

What does ℝ stand for?

State the domain of {(0, 2), (1, 5), (2, 8)}.

y = x² − 9. What is the smallest output?

More practice: on your own

State the domain and range of f(x) = x² + 8x + 20 by completing the square.

A parabola opens down, has vertex (−1, 6) and is drawn only for −4 ≤ x ≤ 2 with closed dots. Its endpoints are (−4, −3) and (2, −3). State the domain and range.

A taxi charges $4 plus $2 per kilometre: C(d) = 2d + 4. Trips are at most 50 km. State a reasonable domain and range.

Bus passes cost $3 each and a class of 28 students needs some. Total cost T(p) = 3p. Is the graph continuous or discrete? State the domain.

Explain why the range of y = (x + 5)² is {y ∈ ℝ | y ≥ 0} even though x can be negative.

Challenge

A rectangular garden has a perimeter of 20 m. If one side is x metres, the area is A(x) = x(10 − x). State the domain and range in context.

A diver jumps from a 10 m platform. Her height is h(t) = −4.9t² + 3t + 10. She enters the water when h = 0. Find the domain (to two decimals) and the range in context.

Write a quadratic function with domain {x ∈ ℝ} and range {y ∈ ℝ | y ≤ −2}, and explain why it works.

Check your understanding

Quiz

What is the range of {(−1, 3), (0, 5), (1, 3), (2, 6)}?

What is the range of y = 3(x − 1)² + 4?

A graph starts at a closed dot (2, 1) and ends at an open dot (7, 4), rising steadily. What is its domain?

How do you read {x ∈ ℝ | x ≠ 0}?

The number of people n on a canoe trip of at most 12 people determines the food cost. Which domain makes sense?

Why is the domain of h(t) = −5t² + 20t + 1 (height of a thrown ball) restricted in context?

For parents and teachers

Big idea: Expectation A1.3 asks students to explain domain and range using numeric, graphical and algebraic representations, to describe them in proper notation, and to explain restrictions that come from real-world contexts.

How to help: Point to a graph (weather, fuel use, a hydro bill chart) and ask, "What x-values does this cover? What y-values?" When your teen writes set notation, have them read it aloud in words; this catches many errors.

Common difficulties:

Swapping domain and range. Domain = inputs = x = horizontal.
Reading the range only from the endpoints and missing a vertex in the middle.
Mixing up ≤ and < at open and closed dots.
Forgetting real-world limits: time and length are not negative, and counts of people or tickets are whole numbers (a discrete graph).
Listing repeated values in a range, such as {4, 1, 0, 1, 4}.