Describe domain and range of linear and quadratic functions

Ontario Grade 11 Mathematics

Quadratic Functions

Use equations, graphs, and context to identify possible inputs and outputs

A function connects an input to an output. For example, a rule might take a number

x

, multiply it by

2

, and add

3

to make an output

y

. The domain tells us which inputs are allowed. The range tells us which outputs the function can produce. These ideas help us describe a function without listing every input-output pair. In this lesson, we focus on linear and quadratic functions. Unless a question gives a restriction or a real-world context, we will treat the input as any real number.

What you will learn

Explain what domain and range mean for a function.
Describe the domain and range of a linear function from its equation or graph.
Describe the domain and range of a quadratic function using its vertex and direction of opening.
Recognize when a context or stated restriction changes the domain or range.

1. Prerequisite bridge: inputs, outputs, and graphs

The variable

x

usually represents an input, and

y

represents the corresponding output. In a function, each allowed input has exactly one output. For instance, in

y=2x+3

, an input of

4

gives an output of

11

.

A graph shows these input-output pairs as points. The horizontal axis shows

x

-values, and the vertical axis shows

y

-values. The domain is the set of

x

-values used by the function. The range is the set of

y

-values produced by the function.

A set is a collection of values. When we say a domain or range is all real numbers, we mean that any number on the number line is included. We can also describe a set using an inequality, such as

y\geq 2

, which means

y

is at least

2

.

\text{domain: }x\text{-values};\quad\text{range: }y\text{-values}

Domain describes inputs, or horizontal values.
Range describes outputs, or vertical values.
A graph can show which values occur even when the equation is not written in a list.

2. Linear functions: follow the line

A linear function has a straight-line graph. A non-horizontal line continues in both directions if no endpoints or restrictions are shown. It therefore uses every real

x

-value. As the line continues, its

y

-values also continue without a largest or smallest value.

For example,

y=2x+3

has domain all real numbers and range all real numbers. Its slope is not zero, so changing

x

changes

y

. The line rises or falls without stopping.

A horizontal line is a special case. In

y=5

, every input gives the same output,

5

. Its domain is all real numbers, but its range contains only

5

. This distinction matters: not every linear function has every real number as an output.

A graph or context can restrict a function. If a question shows only a segment of a line, use only the

x

-values and

y

-values on that segment. In an application, the input may have a meaningful limit, such as time starting at zero. Do not assume such a limit unless the question or context gives it.

y=mx+b

An unrestricted, non-horizontal linear function has all real numbers as both domain and range.
A constant linear function has all real numbers as its domain and one output value as its range.
Use endpoints or context restrictions when they are part of the question.

3. Quadratic functions: use the vertex and opening direction

A quadratic function has a graph shaped like a U or an upside-down U. This graph is called a parabola. Its turning point is the vertex. The vertex gives the lowest or highest output, depending on the direction the parabola opens.

In vertex form,

y=a(x-h)^2+k

, the vertex is

(h,k)

. The squared part,

(x-h)^2

, is never negative. If

a

is positive, the parabola opens upward and

k

is the lowest output. Its range is

y\geq k

. If

a

is negative, it opens downward and

k

is the highest output. Its range is

y\leq k

.

The domain of an unrestricted quadratic function is all real numbers. The input can move left or right without a stopping point on the full graph. The range is different: it starts at the vertex's output and extends upward or downward according to the opening direction.

For example,

y=(x-2)^2+1

has vertex

(2,1)

and opens upward. Its domain is all real numbers, and its range is

y\geq 1

. The input

x=2

gives the lowest output,

1

.

y=a(x-h)^2+k

Read the vertex $(h,k)$ from $y=a(x-h)^2+k$ .
If $a>0$ , the vertex output is the minimum and the range is $y\geq k$ .
If $a<0$ , the vertex output is the maximum and the range is $y\leq k$ .
An unrestricted quadratic function has domain all real numbers.

4. Describing domain and range in words and symbols

You can describe a domain or range in words, with an inequality, or with set notation. An inequality is often the quickest way to show that values begin at a boundary and continue in one direction. For example,

y\geq 1

includes

1

and every larger value.

When using a graph, scan from left to right to describe the domain and from bottom to top to describe the range. Include a boundary value if the graph contains that point. If the graph is a full line or full parabola, its ends continue without a marked endpoint.

Check whether the question describes the full function or only part of it. For a restricted graph, the domain and range come from the part that is actually shown. For a real-world situation, consider which inputs make sense and which outputs are possible. A restriction belongs in the answer only when it is supported by the graph, equation, or context.

x∈\mathbb{R}

Read domain horizontally and range vertically.
Include a minimum or maximum value when the graph reaches the vertex.
Do not add restrictions that the question does not give.

Quick guide for full, unrestricted graphs

Function	Domain	Range
Non-horizontal line	All real numbers	All real numbers
Horizontal line $y=c$	All real numbers	Only $c$
Upward-opening parabola with vertex $(h,k)$	All real numbers	$y\geq k$
Downward-opening parabola with vertex $(h,k)$	All real numbers	$y\leq k$

Worked example

Compare a line and a parabola

Describe the domain and range of the functions

f(x)=3x-2

and

g(x)=-(x+1)^2+4

, assuming each is shown as a full function.

Identify the type of each function
The first rule is linear because it has the form of a straight-line function. Its coefficient of $x$ is not zero, so it is not a horizontal line. The second rule is in vertex form, so its vertex and opening direction can be read directly.
$f(x)=3x-2,\quad g(x)=-(x+1)^2+4$
Describe the line
There is no stated restriction on the input, so the full line uses every real $x$ -value. Since its slope is $3$ , the line is not horizontal and its outputs also extend through all real values.
$D_f=\mathbb{R},\quad R_f=\mathbb{R}$
Find the parabola's vertex and direction
The expression $(x+1)^2$ can be read as $(x-(-1))^2$ . The vertex is therefore $(-1,4)$ . The coefficient of the squared expression is negative, so the parabola opens downward and its vertex gives the greatest output.
$(h,k)=(-1,4),\quad a=-1$
State the parabola's domain and range
The full parabola extends left and right without stopping, so every real input is allowed. Its greatest output is $4$ , and it includes that value at the vertex. All other outputs are less than $4$ .
$D_g=\mathbb{R}, R_g=\{y∈\mathbb{R}\mid y≤ 4\}$

Answer: For

f

, the domain and range are both all real numbers. For

g

, the domain is all real numbers and the range is

y\leq 4

.

Check: The parabola's vertex is included, so

4

belongs to its range. Its downward opening means no output can be greater than

4

.

Common mistakes and how to avoid them

Calling the vertical values the domain and the horizontal values the range.

Correction: Domain is about input

x

-values, read horizontally. Range is about output

y

-values, read vertically.

Saying every linear function has all real numbers as its range.

Correction: A horizontal line has only one output value. A non-horizontal full line has all real numbers as its range.

Using the vertex's

x

-coordinate as the quadratic's minimum or maximum output.

Correction: The vertex is

(h,k)

. Its output is the second coordinate,

k

, which sets the boundary of the range.

Giving an unrestricted full function a restricted domain without a reason.

Correction: Use all real inputs unless a graph, equation, or context states a limit.

Lesson summary

Domain is the set of allowed inputs; range is the set of possible outputs.
For a full non-horizontal line, both domain and range are all real numbers. A horizontal line has one output.
For an unrestricted quadratic, the domain is all real numbers.
The quadratic vertex sets the range boundary: an upward-opening parabola has a minimum, and a downward-opening parabola has a maximum.
Apply restrictions only when the graph, equation, or context provides them.

Check your understanding

Question 1

A full graph is the parabola

y=(x-3)^2-2

. Which statement describes its range?

$y\geq -2$
$y\leq -2$
All real numbers
Only $y=-2$

Show answer and explanation

y\geq -2

The vertex is

(3,-2)

. The positive coefficient means the parabola opens upward, so its minimum output is

-2

and the range is

y\geq -2

.

Question 2

What is the range of the full horizontal line

y=6

?

All real numbers
$y\geq 6$
$y\leq 6$
Only $y=6$

Show answer and explanation

Only

y=6

Every input gives the same output,

6

, so the range contains only that value.

Question 3

A full, non-horizontal linear function has no stated restrictions. What is its domain?

Only positive real numbers
All real numbers
One input value
All real numbers except zero

Show answer and explanation

All real numbers

A full line continues left and right without stopping, so every real input is included.

Key terms

Function: A rule that assigns exactly one output to each allowed input.
Domain: The set of input values allowed for a function.
Range: The set of output values a function can produce.
Linear function: A function whose graph is a straight line.
Quadratic function: A function whose graph is a parabola.
Vertex: The turning point of a parabola; it gives the minimum or maximum output.

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