Learning activity 2.2: Determining domain and range

In Learning Activity 1.1, we discussed the domain and range of a function. In this learning activity, we will study how to represent domain and range in set notation and how to find domain and range from a given graph and an equation.

Let us review the following definitions:

The domain of a function is the set of all values of the independent variable for which the function is defined.

The range of a function is the set of all values of the dependent variable that are determined from the values in the domain.

For real-world problems, the domain and range are often restricted to values that make sense to us.

Equation for the trajectory of a thrown horseshoe

In the last unit you were introduced to a problem involving a player throwing a horseshoe that was modelled by the equation $h = - \frac{1}{50} (3 x^{2} - 37 x - 86)$ , where $h$ is the height in meters and $x$ is the horizontal distance in meters.

What would this equation be in function notation?

$h (x) = - \frac{1}{50} (3 x^{2} - 37 x - 86)$

You were shown the following graph of the situation.

The key points; $x$ -intercepts and vertex are labelled for your reference.

Notebook

In your notebook, answer the following questions about the graph we just reviewed. When you are finished, check your answers with the suggested answers provided.

a) Approximately, for what horizontal distance will the horseshoe reach a height of 3 m?

At $2$ m and $10.2$ m. Be sure to include to include both options when examining the graph.

b) Approximately, for what horizontal distance will the horseshoe reach a height of $4.5$ m?

The maximum height of the horseshoe is about $4$ m, it will never reach a height of $4.5$ m. This point does not exist on the graph.

c) Approximately, what height will the horseshoe be at when the horizontal distance is $4$ m?

Approximately $3.6$ m in height.

d) Approximately, what height will the horseshoe be at when the horizontal distance is $16$ m? Does this value make logical sense?

It will be at a height of about $- 1.6$ m.

This does not make sense because you cannot have a height of $- 1.6$ m.

Notice that not all values exist on the graph. For instance, you cannot find a $y$ -values above $4$ m.

Notice that not all values on the graph would be logical for this problem. For instance, you cannot have a height of $- 1.6$ m.

Set notation

In the ‘Minds on’ activity you learned that only some independent ( $x$ ) and dependent ( $y$ ) values exist in a relation.

The set of acceptable $x$ values in a relation is called the domain.

The set of acceptable $y$ values in a relation is called the range.

Domain

Domain is all the possible $x$ values of a function.

Range

Range is all the possible $y$ values of a function.

Previously you learned that a real number is any number that can be represented as either a fraction or non-terminating, non-repeating decimal.

Domain:

If we examine the equation of area of a circle from the last activity, $A = π r^{2}$ , logically, what does the radius, $r$ , have to be bigger than?

Has to be $0$ or bigger.

It would be impossible to have a negative value as the radius.

Since radius is the independent variable, we can state the domain (D) as follows:

Explaining the meaning of each part of the domain

If this equation did not refer to area of a circle and just represented an equation $y = π x^{2}$ , there would be no restrictions on $x$ and all values would be acceptable. We would write this domain as $D = {x ε R}$ .

Range:

If we examine the equation of area of a circle from earlier, $A = π r^{2}$ , logically, what restrictions would be on the area?

Has to be $0$ or bigger.

It would be impossible to have a negative value as the area.

Since area is the dependent variable, we can state the range ( $R$ ) as:

Graph depicting y=Sqrt of x and a second graph depicting a parabola of y=x^2

If the equation did not refer to area of a circle and just represented an equation $y = π x^{2}$ , there would still be this restriction on the $y$ values because the $x$ values are all squares, resulting in only positive $y$ values. We would write this range as $R = {y ϵ R | y \geq 0} .$

Domain and range from a graph

Explore this!

Let’s explore more about the domain and range of quadratic functions discussed in the following video.

Try it

Let us now practice and determine the domain and range of the following graph.

What is the domain?

The smallest value that corresponds to the input values along the horizontal axis is $x = - 4$ . Therefore, the domain is the set of all real numbers $x$ such that $x$ is greater than or equal to $- 4$ . A shorter way to express the domain is in set notation, $D = {x \in R | x \geq - 4}$ .

What is the range?

The smallest value that corresponds to the output values along the vertical axis is $y = 0$ . Therefore, the range is the set of all real numbers $y$ such that $y$ is greater than or equal to $0$ . In set notation, $R = {y \in R | y \geq 0}$ .

Determine the domain and range of the following graph.

What is the domain?

All the input values along the horizontal axis, in both directions, will be used so the domain is the set of all real numbers, $D = {x \in R}$ .

What is the range?

The smallest value that corresponds to the output values along the vertical axis is $y = - 1$ and the largest value is $y = 1$ . So all the $y$ -values are in between, including $- 1$ and $1$ . In set notation, $R = {y \in R | - 1 \leq y \leq 1}$ .

Domain and range from a linear equation

Examine the following graph of $y = 4 x - 3$ .

Slope of $4$
$y$ -intercept of ( $0, - 3$ )

Are there any restrictions on the $x$ or $y$ values?

No, the $x$ and $y$ value will continue forever in both directions.

State the domain and range for the line.

$D = {x ϵ R} a n d R = {y ϵ R} .$

Domain and range from a quadratic equation

The graph of a quadratic function is a parabola. When looking at a parabola, are there any restrictions on the $x$ values?

No, the $x$ values will continue in both negative and positive direction for all parabolas.

State the domain of any quadratic function.

$D = {x ϵ R}$

Note that this may change when the quadratic represents a ‘real life’ problem.

A parabola may open up (concave up) or may open down (concave down). When the leading coefficient on $x$ is positive, the quadratic is concave up. When the leading coefficient on $x$ is negative, the quadratic is concave down. The range of aquadratic function depends on the direction of opening of the parabola and the minimum or maxim mvalue of the function.

Parabola may open up (concave up for a positive quadratic) or may open down (concave down for a negative quadratic) — Press here for long description(Open in new window)

Parabola (concave up)

The vertex (p,q) is the minimum point when a parabola is concave up.

In this case, $y$ =q is the minimum value and so the range is $R = {y \in R | y \geq q}$ .

Parabola (concave down)

The vertex (p,q) is the maximum point for a parabola when it is concave down.

In this case, $y = q$ is the maximum value and so the range is $R = {y \in R | y \leq q}$ .

Examples

1. Consider the domain and range of the following parabola:

What is the domain and range?

This parabola is concave up.

The vertex $(0, - 4)$ is the lowest point.

The value $y = - 4$ of the vertex is the minimum value of the parabola.

The domain is $D = {x \in R}$ .

The range is $R = {y \in R | y \geq - 4}$ .

2. Consider the domain and range of the following parabola:

What is the domain and range?

This parabola is concave down.

$D = {x \in R}$

$R = {y \in R | y \leq 0}$

Good job. It’s time to consolidate your learning.

Summary

The set of all values of the independent variable is called the domain. On a Cartesian graph, the domain is the set of all possible values of the independent variable $x$ . Example: The relation $y = V x$ . exists only for positive values of $x$ . The domain of this relation is $x = 0$ and all positive values of $x$ .
The set of all values of the dependent variable is called the range. On a Cartesian graph, the range is the set of all possible values of the dependent variable $y$ . Example: $y = x^{2}$ The range is $y = 0$ and all positive values of $y$ .
A function can also be defined as a relation in which each element of the domain corresponds to only one element of the range.
The vertical-line test can be used to check whether the graph of a relation represents a function. If two or more points lie on the same vertical line, then the relation is not a function.
It is often necessary to define the domain and range using set notation. For example, the set of numbers ... , -2, -1, 0, 1, 2, ... is the set of integers and can be written in set notation as ${x | x \in I}$ , where the symbol "|" means "such that" and the symbol "∈" means "belongs to" or "is a member of." So set notation for the integers would be read as "The set of all $x -values$ such that $x$ belongs to the integers."

Self-check

As a self-directed learner, you will be reflecting on your learning process and checking your understanding in order plan for success. Make a note of your understanding of the success criteria from today’s activity.

Make a note of your understanding of the success criteria from today’s activity.

Rate your understanding on a scale of five to one.

Five means “I have a thorough understanding.” One means “I am confused.”

Are you able to

Statement	1	2	3	4	5
Use set notation to describe the domain and range of a graph
Determine the domain and range from a graph
Determine the domain and range from a linear equation
Determine the domain and range from a quadratic equation

If you feel you need more practice visualizing when solve quadratic equations, you may want to have Geogebra (Opens in new window) out to see the relation on a graph to aid in your algebraic solution.

Math Journal

At the end of the course, you will fine-tune 8 entries (two from each unit) from your math journal and submit them as your “Culminating Assessment - Math Journal” (Opens in new window)

Summarize how you can determine the range based on the concavity of the parabola. Your summary may look like this:

Parabola	Vertex (p, q)	Domain	Range	Suggested Answers
Concave up	Minimum point	${x \in R}$		${y \in R \| y \geq q}$
Concave down	Maximum point	${x \in R}$		${y \in R \| y \leq q}$

Press the start button if you wish to use the GeoGebra graphing tool. The GeoGebra tool will open in a new window, and can be used throughout this part of the learning activity.

Start

Again, in your math journal summarize the meanings of the domain and range of a graph, how to determine domain and range from a graph vs. from an equation and how domain and range can change when describing a real-life situation. Your summary may resemble the following table :

	What is it?	How to determine from a graph?	How to determine from an equation	How does it change for a real-life problem?
Domain
Range

Once you feel comfortable with the success criteria, complete the following questions to assess your progress.

Further practice

Determine the domain and range of the following graph of a function.

$D = {x \in R}$ $R = {y \in R | y \geq - 8}$

Determine the domain and range of the following graph of a function.

$D = {x \in R}$ $R = {y \in R | y \leq 20}$

Making connections

Examine the function, $h (t) = - 0.5 t^{2} + 20 t$ . This function represented the height of a rocket, in metres, after $t$ seconds.

The graph of this function is displayed beside:

Answer the following questions based on the graph in your notebook, and when complete compare your answers with the suggested answers provided.

What physical factors will restrict the domain and range of the quadratic function?

The quadratic function represents the height of a rocket after it is launched.

The rocket is launched at time $t = 0$ s from a height of $0$ m. Once the rocket reaches its maximum height, it will fall to the ground because of gravity. This will determine the range of the function.

Since time cannot be negative, the domain will be a set of positive values beginning at $0$ and ending at the time when the rocket lands.

Determine the domain of this function. How long was the rocket in the air?

The domain of this function is $D = {t \in R | 0 \leq t \leq 40}$ . The rocket was in the air for $40$ s.

Determine the range of this function. What was the maximum height reached by the rocket?

The range is $R = {h \in R | 0 \leq h \leq 200}$ .

The maximum height reached by the rocket was $200$ m.