Learning activity 2.1: Investigating function notation

In the previous unit, you learned about the definition of a function and how a function is different from a relation.

In this learning activity, you will learn about the properties of a function and how to present functions in various ways. We will also learn about function notation and the domain and the range of a function. Bringing this all together, we will discuss some real word situations that can be modelled by a table of values and by graphs.

Representing real-world applications of functions

Try it!

Graphs can be used to represent real-world applications of functions. In this learning activity, you will explore the relationship between real-world scenarios and how they can be rendered in graph form. Consider the following images, in each there is an object in motion. Try to imagine the path that the object in motion would take. From your imagined motion of the object, sketch a graph that represents that motion over time. Consider the values you would use for the x-axis and y-axis in your graphs.

Example 1: A football in the air after being kicked.

Example 2: The movement of a single carriage on a ferris wheel.

Example 3: A person in a bumper car moving around.

Join the discussion

Share a sketch of your graph and the values you would use for the x-axis and y-axis with your classmates. Does your graph look similar to your classmates?

Press the “Join The Discussion” button when you’re ready to engage.

Now that you’ve considered how these different paths of motion would be represented graphically you will explore how different variables relate to each other and how these relationships can be represented and investigated mathematically.

Defining a function

Mathematical problems arise from real-world situations that involve relationships between quantities. The following are some examples:

To determine the area of a circular patio, you need to know its radius.
To find the revenue earned from selling tickets that cost $5 each, you need to know the number of tickets sold.

Mathematical relations are represented by equations that contain appropriate variables. For example:

The equation for the area $A$ of a circle with radius $r$ is $A = π r^{2} .$
The equation for the revenue $R$ earned by selling $t$ tickets that cost $5 each is $R = 5 t$ .

In each equation, one quantity depends on the other. Therefore, one quantity is called the dependent variable (which is plotted on the vertical axis of a graph) and the other quantity is the independent variable (which is plotted on the horizontal axis of a graph).

In the equation $A = π r^{2} .$ , the area depends on the radius, so $A$ is the dependent variable and $r$ is the independent variable. In the equation $R = 5 t$ , the revenue $R$ depends on the number of tickets sold, so $R$ is the dependent variable and $t$ is the independent variable.

The following are tables of values for each of the relations discussed previously. The independent variable (first column in both tables) corresponds to the input value (on the horizontal axis). The dependent variable (second column in both tables) corresponds to the output value (on the vertical axis).

Table of values for $A = π r^{2}$

$r$	$A = π r^{2}$
0	0
1	3.14
2	12.57
3	28.27
4	50.27
5	78.53
6	113.10
7	153.94
8	201.06
9	254.47
10	314.16

Table of values for $R = 5 t$

$t$	$R = 5 t$
0	0
1	5
2	10
3	15
4	20
5	25
6	30
7	35
8	40
9	45
10	50

Notice that in both tables, each input value produces only one output value.

This type of relation is called a function; a relation where an input value produces only one output value. In a table, that means you would never see the same input value in two or more rows. Each input value only appears once.

The values in each of the preceding tables can be represented as ordered pairs. The first coordinate is the input (independent) value and the second coordinate is the output (dependent) value.

Notebook

In your notebook, work through the following questions. Check your work with the suggest answers provided.

Indicate the values from the table in the second function $R = 5 t$ as coordinate points.

$(0$ , $0)$ , $(1$ , $5)$ , $(2$ , $10)$ , $(3$ , $15)$ , $(4$ , $20)$ , $(5$ , $25)$ , $(6$ , $30)$ , $(7$ , $35)$ , $(8$ , $40)$ , $(9$ , $45)$ , $(10$ , $50)$

What do you notice about how many output ( $y$ values) there are for a given input value ( $x$ value).

Note that none of these ordered pairs have the same first value with different second values. This is true only for relations that are functions. This follows the idea that functions have one output ( $y$ value) for a given input ( $x$ value).

Is the relation with the following coordinate points a function? $(9, - 3)$ , $(4, - 2)$ , $(1, - 1)$ , $(0, 0)$ , $(1, 1)$ , $(4, 2)$ , $(9, 3)$ why or why not?

It is not a function because the ordered pairs $(9, - 3)$ and $(9, 3)$ have the same first coordinate as do the ordered pairs $(4, - 2)$ , $(4, 2)$ and also $(1, - 1)$ , $(1, 1)$ . Therefore, the input values $9$ , $4$ , and $1$ each have two output values.

Considering the degree of both $A = π r^{2}$ and $R = 5 t$ , what specific type of relation/function do these represent?

The relation $A = π r^{2}$ is a quadratic function because a variable is squared (which means the degree of the equation is two). The relation $R = 5 t$ is a linear function because the degree of the equation is one.

Please note that all linear relations that can be represented by the form $y = m x + b$ , where $m$ is the slope and $b$ is the intercept of the equation.

The following are a few examples of Linear Functions:

$y = 5 x + 3$ , Here $m = 5$ and $b = 3$
$y = - x$ , Here $m = - 1$ and $b = 0$
$y = 2 x - 2$ , Here $m = 2$ and $b = - 2$
$y = - 6 x - 1$ , Here $m = - 6$ and $b = - 1$

All quadratic relation that can be represented by the form, $y = a x^{2} + b x + c$ , where $a$ , $b$ and $c$ are constants with $a \neq 0$ .

The following are a few examples of Quadratic Functions:

$y = x^{2} + 2 x + 1$
$y = x^{2} + 5$
$y = - 9 x^{2} + 3 x$
$y = 8 x^{2} - x + 7$

Determining if a relation is a function from an equation:

The equation of a relation does not represent a function if an input value can be found that produces more than one output value. As you have just learned, linear equations (except for vertical lines) and quadratic equations are functions. For equations you are unsure of, you must test $x$ -values that include zero and negative numbers.

Think

Which of the following relations represent functions? Can you justify your answer? Compare your solutions to the suggested answers provided.

$3 x + 4 y = 9$

$3 x + 4 y = 9$ is a linear equation and so it is a function.

$x^{2} + y^{2} = 36$

The equation $x^{2} + y^{2} = 36$ is neither linear nor quadratic.

Find a value of $x$ that may produce more than one $y$ -value.

Substitute $x = 0$ into the equation $x^{2} + y^{2} = 36$ and solve for $y$ .

${(0)}^{2} + y^{2} = 36$

$y^{2} = 36$

$y = \pm \sqrt{36}$

$y = \pm 6$

When $x = 0$ , there are two values of $y$ . There are two output values for one input value.

This is not a function.

$A = s^{2}$

$A = s^{2}$ is a quadratic relation, so it is a function.

$(1, 2), (3, 9), (1, 5), (4, 10)$

$(1, 2), (3, 9), (1, 5), (4, 10)$ is not a function because the points $(1, 2)$ and $(1, 5)$ have the same $x$ -coordinates and different $y$ -coordinates.

Determining if a relation is a function from a graph (vertical line test)

You have learned how to determine if a relation is a function from a set of ordered pairs and from an equation. Now you will learn how to distinguish if a relation is a function from a graph.

The graph of each function $A = π r^{2}$ and $R = 5 t$ is created by plotting the input values along the horizontal axis and the output values along the vertical axis. Examine the graphs of these two functions in the following:

Area of a circle

The graph is half of a parabola since only positive values can be used for the radius.

Revenue from ticket sales

The graph is a straight line and only positive integer values can be used for the number of tickets.

How do you determine if it’s a function?

Suppose you are given the graph of a relation. How can you tell from the graph if the relation is a function? To determine if a relation is a function from a given graph, use the vertical line test. This test checks for any points (ordered pairs) on the graph that may contain the same input value (first coordinate). You are about to learn what the vertical line test is in great detail.

Interactive vertical line test

Now let's use the vertical line test in action!

The graph of a relation is a function if no two points on the graph can be connected by any vertical lines.

Slide the vertical line across the function to determine if the curve is a function or not.

If the line only touches the graphed line or curve one point at a time then it’s a function.
If at any point the line touches more than one point then it’s not a function.

In order to communicate clearly that you are representing a function, you must follow function notation. In this course, the function will usually start off with the “value of $f$ at $x$ ” or simply “ $f (x)$ .”

Start

Function notation

You may have noticed that the relations and functions from earlier in this learning activity have been represented as:

ordered pairs
a table of values
a graph
an equation

The defining equation of any function can also be represented by function notation which you will next explore in more detail. The following table demonstrates how to express the defining equation of a linear function and a quadratic function using the function notation.

Defining equation	Function notation
$y = 3 x - 7$	$f (x) = 3 x - 7$
$y = 2 x^{2} - 4 x + 5$	$f (x) = 2 x^{2} - 4 x + 5$

The symbol $f (x)$ is read as the “value of $f$ at $x$ ” or sometimes “ $f$ of $x$ .” This symbol simply means that the expression that follows the equal sign is a function and contains the variable “ $x$ .”

Letters other than $f$ and $x$ can be used for function notation.

For instance, if the equation $h = - 0.5 t^{2} + 40 t$ represents the height, $h$ , of a ball after time, $t$ , seconds, then it is appropriate to use the function notation $h (t) = - 0.5 t^{2} + 40 t$ . The variable inside the brackets represents the input value.

Complete the chart for each of the equations in the first column of the following table:

Defining equation	Function notation
$y = 2 x + 9$	$f (x) = 2 x + 9$ Notice that we use ‘ $f$ ’ to replace the ‘ $y$ ’ for function notation.
$P = 6 t^{2} - 9 t + 3$	$P (t) = 6 t^{2} - 9 t + 3$
$V = 8 w^{3}$	$V (w) = 8 w^{3}$
$A = π r^{2}$	$A (r) = {π r}^{2}$

Using function notation like a machine

As you have learned, when using function notation, the variable inside the brackets represents the input values. For example, to determine $f (2)$ for the function $f (x) = 3 x - 7$ , we use the input value of $2$ , $x = 2$ .

$f (x) = 3 x - 7$

$f (2) = 3 (2) - 7$

$f (2) = - 1$

The preceding example shows that when the input value is $2$ , the output value is $- 1$ . This is represented by the point $(2, - 1)$ .

You can think of a function as a machine that performs operations on numbers. Input numbers go into the machine and have operations (adding, subtracting, multiplying, etc) performed on them. When all the operations are completed, they come out of the machine as output values.

The input/output machine

The following is the I/O machine in action!

Start

You can examine that there is a direct connection between the input and output of a function.

In some cases, you do not know the equation of a function or it is not given. Function notation can also be used when a defining equation is not known or not given. Try the following exercise.

The following is the graph of $y = g (x)$ .

Use the graph to determine $g (2)$ , $g (0)$ , $g (- 1)$ , and $g (- 3)$ . State the ordered pair for each.

Examine the graph of $y = g (x)$ again.

Does the graph represent a function? Justify your answer.

Yes, the graph represents a function because it passes the vertical line test.

Some examples of functions in everyday life

In our surroundings, we come across many examples of functions.

Distance travelled by a continuously moving object is a function, as the moving object will be at a unique distance at a particular time. Moving objects cannot be at different distances at a given time.

Person on a bicycle speeds through traffic on Stockholm’s Zebra Crossing.

You can find another example of a function in a grocery store because a unique price is associated to a particular brand of product. There cannot be two different prices of a given brand of product.

Self-check

Take a moment to reflect on what you have learned about investigating function notation in this learning activity.

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.

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
Determine if a relation is a function from an equation
Determine if a relation is a function from a graph
Determine the function’s value given an $x$ value algebraically using function notation
Determine the function’s value given an $x$ value graphically using function notation

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 “ Assessment - Math Journal” (Opens in new window).

In your math journal compare a relation and a function. Your summary may resemble the following table:

Comparing a relation and a function

Similarities between a relation and a function	Differences between a relation and a function

List the steps you would take to determine the function’s value (output) given an $x$ value (input) in your own words.

Portfolio

Submit your answers to your portfolio for feedback

Given fx=4x2−2x+3 determine the following using your notebook.
1. Determine $f (12)$
2. State the ordered pair that corresponds to the input value of $12$
3. Determine $f (- 2)$
4. State the ordered pair that corresponds to the input value of $- 2$
The function $h (t) = - 0.5 t^{2} + 20 t$ represents the height, $h$ in metres, of a toy rocket, $t$ seconds after it is launched into the air. Determine the height of the rocket after $t = 4$ s.

Submit your portfolio item(s) by pressing the “Go To Portfolio” button.

Connections

Once you feel ready, try the following questions that connect all topics from this activity. Here is an opportunity to test your understanding of function notation.

Complete a chart like this one for each equation in the first column. Check your answers when you are done.

Equation	Function notation
$y = - 6 x + 5$	$f (x) = - 6 x + 5$
$C = 2 v^{2} - 20 v + 5.3$	$C (v) = 2 v^{2} - 20 v + 5.3$
$A = 2 w (60 - w)$	$A (w) = 2 w (60 - w)$
$h = - 4.9 t^{2} + 25 t + 9$	$h (t) = - 4.9 t^{2} + 25 t + 9$

For each of the following functions, determine the output value for each given input value and then state the corresponding ordered pair.

$g (x) = 9 x + 5$ , $x = - \frac{2}{3}$

The output value is $- 1$ .

The ordered pair is $(- \frac{2}{3}, - 1)$ .

$f (x) = 3 x^{2} - 4 x + 6, x = - 5$

The output value is $101$ .

The ordered pair is $(- 5, 101)$ .

Representing real-world applications of functions

Try it!

Join the discussion

Defining a function

Table of values for A=πr2

Table of values for R=5t

Notebook

Determining if a relation is a function from an equation:

Think

Determining if a relation is a function from a graph (vertical line test)

Interactive vertical line test

Function notation

Using function notation like a machine

The input/output machine

Some examples of functions in everyday life

Self-check

Math journal

Comparing a relation and a function

Try it!

Portfolio

Connections

Table of values for $A = π r^{2}$

Table of values for $R = 5 t$