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A. Quadratic Functions · Quadratic Functions

Domain and Range of Linear and Quadratic Functions

A2.3About 65 minutes
Learning goals
  • I can explain what the domain and range of a function are.
  • I can find the domain and range from ordered pairs, mapping diagrams, graphs and equations.
  • I can write domain and range using lists and set notation.
  • I can use the vertex to find the range of a quadratic function.

Words to know

domain
The set of all possible inputs (x-values) of a function.
range
The set of all outputs (y-values) a function produces.
set notation
A way to describe a set, such as {x ∈ ℝ | x ≥ 3}: all real x such that x is at least 3.
ℝ
The set of real numbers.
∈
"Is an element of" (belongs to).
closed dot / open dot
On a graph, an endpoint that is included / not included.

What can go in, what can come out

A phone battery reading can be any value from 0% to 100%. It can never be 130% or −5%. The number of people on a bus is a whole number, never 23.7. Every function has a set of inputs that make sense and a set of outputs it can actually produce. These sets are called the domain and the range.

Definitions

  • The domain of a function is the set of all possible inputs (x-values).
  • The range is the set of all outputs (y-values) the function actually produces.

How to write them. For a short list, use braces: {−3, −1, 0, 2}. For an interval of real numbers, use set notation:

  • {x ∈ ℝ} means "x is any real number". (∈ means "is an element of", and ℝ is the set of real numbers.)
  • {x ∈ ℝ | x ≥ 3} means "all real numbers x such that x is greater than or equal to 3". The bar | is read "such that".
  • {y ∈ ℝ | −2 < y ≤ 5} means "all real y greater than −2 and less than or equal to 5".

On graphs, a closed (filled) dot means the endpoint is included (≤ or ≥). An open dot means it is not (< or >).

Worked example

Example 1: From ordered pairs

State the domain and range of {(−3, 2), (−1, 4), (0, 2), (2, 5)}.

  1. The domain is the set of first coordinates: −3, −1, 0, 2.
  2. The range is the set of second coordinates: 2, 4, 2, 5. List each value once, usually in order.
  3. D = {−3, −1, 0, 2} and R = {2, 4, 5}.

D = {−3, −1, 0, 2}; R = {2, 4, 5}.

Reading domain and range from a graph

Imagine a light shining straight down on the graph. Its shadow on the x-axis is the domain. Now shine the light from the side: the shadow on the y-axis is the range.

Arrows on a graph mean it keeps going forever in that direction.

xy−4−3−2−1012345−10123456789(−2, 1) included(3, 6) not includeddomain: −2 ≤ x < 3range: 1 ≤ y < 6
The shadow of this segment on the x-axis runs from −2 (included) to 3 (not included), and its shadow on the y-axis runs from 1 (included) to 6 (not included). The open dot at (3, 6) means that point is not part of the graph.

Domain and range of linear and quadratic functions

Linear functions f(x) = mx + b:

  • Domain: {x ∈ ℝ}.
  • Range: {y ∈ ℝ} when m ≠ 0. A horizontal line f(x) = b has range {b}, a single value.

Quadratic functions f(x) = ax² + bx + c:

  • Domain: always {x ∈ ℝ}. You can square any number.
  • Range: decided by the vertex (h, k) and the direction of opening.
  • Opens up (a > 0): k is the minimum, so R = {y ∈ ℝ | y ≥ k}.
  • Opens down (a < 0): k is the maximum, so R = {y ∈ ℝ | y ≤ k}.

So to find the range of a quadratic, find the vertex first. Vertex form f(x) = a(x − h)² + k shows it directly. If you know the zeros, the vertex is halfway between them.

xy−4−3−2−10123456−5−4−3−2−10123456vertex (1, −4)range: y ≥ −4domain: all real x
For f(x) = (x − 1)² − 4, the parabola spreads left and right forever, so the domain is every real number. Its lowest point is the vertex (1, −4), so the range is y ≥ −4.
Worked example

Example 2: Quadratics in different forms

State the domain and range of each function.

(a) f(x) = −2(x − 3)² + 5

(b) g(x) = (x − 1)(x − 7)

(c) h(x) = 0.5x² − 8

  1. Every quadratic has domain {x ∈ ℝ}. The work is in the range.
  2. (a) Vertex form: vertex (3, 5) and a = −2 < 0, so the parabola opens down and 5 is the maximum. R = {y ∈ ℝ | y ≤ 5}.
  3. (b) Zeros 1 and 7, so the axis of symmetry is x = (1 + 7) ÷ 2 = 4. g(4) = (3)(−3) = −9. Expanding gives x² − 8x + 7, so a = 1 > 0 and the parabola opens up. R = {y ∈ ℝ | y ≥ −9}.
  4. (c) h(x) = 0.5(x − 0)² − 8 has vertex (0, −8) and opens up. R = {y ∈ ℝ | y ≥ −8}.

Domain {x ∈ ℝ} for all three. Ranges: (a) y ≤ 5, (b) y ≥ −9, (c) y ≥ −8.

Worked example

Example 3: Part of a parabola

Only part of y = x² is drawn: from x = −1 to x = 3, with both endpoints included. State the domain and range.

  1. Domain: the x-values run from −1 to 3. D = {x ∈ ℝ | −1 ≤ x ≤ 3}.
  2. For the range, do not just use the endpoints. The endpoints give y = (−1)² = 1 and y = 3² = 9.
  3. But the vertex (0, 0) is inside this domain, so the graph dips down to y = 0.
  4. R = {y ∈ ℝ | 0 ≤ y ≤ 9}.

D = {x ∈ ℝ | −1 ≤ x ≤ 3}; R = {y ∈ ℝ | 0 ≤ y ≤ 9}.

The vertex trap

When a graph is only part of a parabola, check whether the vertex is inside the domain. If it is, the vertex value is one end of the range, not the y-value at an endpoint.

Just for fun

Why did the function refuse an invitation to x = 12?

It checked its calendar and that date was not in its domain.

Hands-on activity

Domain and range from a weather forecast

You need: a weather app or a newspaper forecast that shows hourly temperatures for one day, graph paper and a pencil.

  1. Copy the hourly forecast for the next 12 hours into a table: time (hours from now, 0 to 11) and temperature (°C).
  2. Is temperature a function of time? Explain in one sentence.
  3. Plot the points and join them with a smooth curve.
  4. Write the domain in set notation (it is the time interval you used).
  5. Write the range in set notation, from the lowest to the highest temperature on your graph.
  6. Repeat with a different relation from your home, such as the phone battery percentage every hour from when you wake up.

What to look for: temperature is a function of time (one temperature at each moment). For hours 0 to 11, the domain is {t ∈ ℝ | 0 ≤ t ≤ 11}. The range runs from the lowest forecast temperature to the highest, for example {T ∈ ℝ | −4 ≤ T ≤ 2} on a cold December day.

Your turn

1

State the domain and range of {(1, 5), (2, 7), (3, 5), (4, 9)}.

2

State the domain and range of f(x) = x² + 3.

3

State the range of g(x) = −(x + 2)² + 6.

4

State the domain and range of h(x) = 2x − 7.

5

A graph is a segment from (−1, 2) to (4, 12), with both endpoints included. State its domain and range.

6

State the range of f(x) = (x − 2)(x − 8).

7

Write {x ∈ ℝ | x > −3} in words.

More practice: warm-up

1

The domain is the set of all ______; the range is the set of all ______.

2

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

3

State the range of f(x) = x².

4

State the range of f(x) = 7.

5

Write "all real numbers less than or equal to 4" in set notation.

More practice: on your own

1

State the range of f(x) = 3(x − 1)² − 12. Does the graph cross the x-axis?

2

State the range of f(x) = −0.5x² + 4x.

3

A mapping diagram has arrows 2 → 5, 3 → 5 and 4 → 6. State the domain and range.

4

The graph of y = x² is drawn only for −2 ≤ x ≤ 4 (endpoints included). State the domain and range.

5

Do f(x) = x² + 1 and g(x) = (x − 4)² + 1 have the same range? Explain.

Challenge

1

Write a quadratic function with range {y ∈ ℝ | y ≤ −2}. Explain why it has no zeros.

2

Find the range of f(x) = x² − 6x + 5.

3

f(x) = −(x − 1)² + 4 is drawn only for 2 ≤ x ≤ 5. State the range. Why is the maximum not 4?

Check your understanding

Quiz

1.

What is the range of f(x) = (x − 3)² − 7?

2.

What is the domain of every quadratic function f(x) = ax² + bx + c?

3.

State the domain of {(2, 3), (4, 3), (6, 1)}.

4.

What is the range of f(x) = −2(x + 1)² + 8?

5.

What does {x ∈ ℝ | x < 5} mean?

6.

An open dot at the end of a graph means…

For parents and teachers

Big idea: expectation A2.3 asks students to explain the meaning of domain and range using numeric, graphical and algebraic representations of linear and quadratic functions, and to describe them appropriately (here, with lists and set notation).

How to help: point at a graph or chart in a news story or app (a temperature forecast, a stock price) and ask "What is the domain? What is the range?" Ask your teen to read set notation aloud in words.

Common difficulties:

  • Mixing up domain (x) and range (y).
  • Using the endpoints of a partial parabola for the range and missing the vertex in between.
  • Getting the inequality direction wrong for parabolas that open down (y ≤ k, not y ≥ k).
  • Forgetting that open dots mean < or >, not ≤ or ≥.