5.6 Functions: Part 1

Leanne Thompson

5.6 Functions: Part 1

In this lesson, we will begin to look at a type of relation called a function. You will learn to identify when a relation can be classified as a function and when it cannot.

A relation is considered to be a function when there will be only one y output for any single x input. If for at least one of the inputs (x values) there is more than one output (y value), the relation is not considered to be a function.

Example 1

Table 5.6.1
Function	Not a Function
This relation is a function because for each input (x value), there is only one output (y value).	This relation is not a function because for at least one of the inputs (x values), there is more than one output (y value). In this example, the input of x = –3 has two outputs: y = 1 and y = 4.

Practice 1

Determine which of the following relations are functions. Explain why or why not.

a) (2, 3), (4, 7), (1, −5), (6, 0), (−3, 8)

b) (−3, 4), (−2, 5), (−3, −6), (1, 7), (−3, 5)

c)

Graph with points (-5,1). (-4,1), (-3,1), (-1,4), (-1,1), (-4,-4), and (-4,2)

d)

Graph with points (-2,-2) (-1,-1) (0,0) and (1,1)

e)

Circle graph x spans -2 to 2 and y spans -2 to 2.

f)

Parabola shaped (u-shape) graph opening up with x-ints at (-1,0) and (1,0) and y-int at (0,1)

g)

A graph of the equation y = 1 third x−1.

h)

Horizontal line passing through y=-2.

The Vertical Line Test

The vertical line test is a method that can be used to determine whether or not a graph represents a function.

If every vertical line intersects the graph at no more than one point, then the relation is a function.
If any vertical line drawn on the graph intersects the graph at more than one point, then the relation in not function.

Practice 2

Use the vertical line test to verify your answers in Practice 1 (c–h).

Practice 3

Which of the following real-life situations represents a function?

a) The relationship between a book’s title and its author.
b) The relationship between a person’s favourite colour and their number of siblings.
c) The relationship between a teacher’s name and the courses they teach.
d) The relationship between a dog’s breed and its weight.

Function Notation

If a relation is a function, we can write the equation of the relation in function notation. To write an equation in function notation, substitute the y in the equation with f(x). Notice in the examples below, other letters besides f can also be used to denote a function. If you are working with more than one function, using different letters to represent each function will allow you to distinguish between the functions.

Example 2

Table 5.6.2
Equation of the Graph	Function Notation
$y = 3x-5$	$f(x) = 3x-5$
$y = 2x^2-3x+2$	$f(x) = 2x^2-3x+2$
$y = -5x+10$	$p(x) = -5x+10$
$y = 3x$	$H(x) = 3x$

Practice 4

Write the following equations in function notation.

a) $y = 2x + 5$

b) $y=-\frac{1}{3}x - 1$

c) $y = x^2 - 3x + 4$

Evaluating for an Output

To evaluate a function, find the output (y value) of the function for a specific input (x value).

Example 3

Evaluate the function $f(x)=3x-5$ for f(4).

Table 5.6.3
Steps	Solution
Identify the x value that you will be substituting into the equation. The x value will be the value inside the bracket beside the letter.	We are evaluating f(4), so x = 4.
Substitute the determined x value into the function, then calculate.	$f(x)=3x-5$ $f(4)=3(4)-5$ $f(4)=12-5$ $f(4)=7$ This tells us that when x = 4, y = 7.

Practice 5

Consider the functions, then evaluate for the given values.

a) $f(x) = 2x + 5$ for $f(3)$

b) $p(x) = 4x - 6$ for $p(-7)$

c) $G(x) = x^2 - 3x + 4$ for $G(3)$

d) $h(x) = -x + 7$ for $h(-2)$

e) $Q(x) = 3x^2 - 2x$ for $Q(1)$

f) $r(x) = x^2 - 2$ for $r(-5)$

Homework

Determine which of the following relations are functions. Explain why or why not.

a) (–1, 2), (2, –3), (3, 4)
b) (0, 0), (–1, 2), (2, –3), (3, 4), (–4, 5)

c) (1, 2), (1, 3), (3, 4), (1, 4), (2, 10)
d) (5, –10), (–6, 11), (7, –12), (–8, 13)

e) (–4, 5), (5, –6), (–4, 7)
f) (2, –3), (3, 4), (2, 5), (–4, 6)

g)

h)

i)

j)

k)

l)

m)

n)

o)

p)

q)

r)
How does the vertical line test help determine if a relation is a function?
Write the following equations in function notation.

a) $y = -6x - 1$

b) $y = x^2$

c) $y = 4^x$

d) $y = 25x$

e) $y = \frac{2}{5}x + 10$

f) $y = x^2 + 10x - 3$
Consider the functions, then evaluate for the given values.

a) $f(x) = 2x + 5$ for $f(3)$

b) $G(x) = -3x + 4$ for $G(2)$

c) $h(x) = x^2 + 2x - 1$ for $h(4)$

d) $P(x) = 5x + 6$ for $P(-2)$

e) $Q(x) = -x^2 - 3x + 1$ for $Q(3)$

f) $r(x) = 4x - 7$ for $r(0)$

g) $S(x) = x^2 + 3x + 2$ for $S(1)$

h) $T(x) = -2x^2 - 3x + 4$ for $T(-1)$

i) $U(x) = 3^x + 1$ for $U(2)$

j) $V(x) = -2x^2 - 5x + 7$ for $V(5)$

k) $w(x) = 5x^2 - x + 3$ for $w(-1)$

l) $Y(x) = 2^x + 3$ for $Y(4)$
Is $y=2x+4$ a function? Explain how you know.
If you graph a horizontal line, is it considered to be a function? Why or why not?
If you graph a vertical line, is it considered to be a function? Why or why not?

Which of the following statements is always true?

a) A graph that is a straight line is always a function.
b) A function assigns exactly one input value for each output value.
c) A function can assign multiple output values for the same input.
d) A function assigns exactly one output value for each input value.

Which of the following sets of ordered pairs are functions? Circle all that apply.

a) (1, 2), (2, 3), (3, 4)
b) (–1, 5), (0, 6), (2, 7), (3, 8)
c) (4, –2), (5, 0), (6, 1)
d) (1, 2), (1, 3), (2, 4)

Which of the following real-life situations represents a function?

a) The relationship between a person’s birth year and their age.
b) The relationship between a student’s name and their grades in all subjects.
c) The relationship between a car’s make and its colour.
d) The relationship between a student’s ID number and their class schedule.

Which of the following real-life situations represents a function?

a) The relationship between a person’s phone number and their home address.
b) The relationship between a city’s population and its area code.
c) The relationship between a person’s height and their shoe size.
d) The relationship between a car’s model and its owner.

Answers

1.

a) function	b) function	c) not a function	d) function	e) not a function	f) not a function
g) function	h) not function	i) function	j) function	k) function	l) function
m) not a function	n) function	o) not a function	p) function	q) not a function	r) function

2. If every vertical line intersects the graph at no more than one point, then the relation is a function. If any vertical line drawn on the graph intersects the graph at more than one point, then the relation is not a function.

3.

a) $f(x) = -6x - 1$	b) $f(x) = x^2$	c) $f(x) = 4^x$
d) $f(x) = 25x$	e) $f(x)=\frac{2}{5}x+10$	f) $f(x) = x^2 + 10x - 3$

4.

a) 11	b) –2	c) 23	d) –4	e) –17	f) –7
g) 6	h) 5	i) 10	j) –68	k) 9	l) 19

5. y = 2x + 4 is a function. If you graphed it, you would find it to be a diagonal straight-line graph going upwards. The graph would have at most one output for each input, making it a function.

6. A horizontal line would be a function because the graph would have at most one output for each input.

7. A vertical line would not be a function because at the vertical, there would be multiple outputs for the same input.

8. d

9. a, b, and c

10. d

11. a

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resources:

Berg, T. (2020). Intermediate algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY-NC-SA 4.0

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

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Example 1

Practice 1

The Vertical Line Test

Practice 2

Practice 3

Function Notation

Example 2

Practice 4

Evaluating for an Output

Example 3

Practice 5

Homework

Answers

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