5.7 Functions: Part 2

Leanne Thompson

5.7 Functions: Part 2

Evaluating for an Output

Practice 1

Evaluate each of the following expressions given the function $g(x)=5x^2-3$ .

a) $g(1)$

b) $g(-5)$

c) $g\left(\frac{1}{2}\right)$

Some questions will require you to work with more than one function. You may need to add the functions or subtract them as seen in the next example.

Example 1

Consider the functions below. Evaluate for $f(2) + g(2)$ .

$f(x) = x^2 - 3$
$g(x) = 2x^2 + 3x$

Table 5.7.1
Steps	Solution
Substitute the expressions for each function.	$f(2) + g(2)$ $=[x^2-3]+[2x^2+3x]$
Substitute the determined x value into the function.	$=[(2)^2-3]+[2(2)^2+3(2)]$
Calculate.	$=4-3+8+6$ $=15$

Practice 2

Consider the functions below. Evaluate for the given values.

$p(x)=x^2-3$
$q(x)=2x+6$
$r(x)=-3x$

a) $q(10)$

b) $p(2) + q(4)$

c) $r(5) + p(-4)$

d) $p(-1) + q(0) - r(1)$

Practice 3

Given the functions below, write a simplified expression for each of the following expressions.

$f(x)=4x-2$
$p(x)=2x^2+3x+5$

a) $f(x+5)$

b) $p(3x)$

c) $p(x+1)$

Evaluating for an Input

For the questions above you were given the value of the input, x, to find the value of the output, f(x). We will now look at questions where you are given the output, f(x), to find the input, x.

Example 2

Consider the function $f(x) = 3x - 5$ . If $f(x) = 10$ , what is the value of x?

Table 5.7.2
Steps	Solution
Substitute the determined f(x) value into the function.	$f(x) = 3x - 5$ $10 = 3x - 5$
Solve for x.	$15 = 3x$ $5 = x$

Practice 4

Solve for x.

a) $g(x) = 4x + 2$ for $g(x) = 14$

b) $h(x) = 5x - 7$ for $h(x) = 8$

c) $p(x) = x^2 - 4$ for $p(x) = 0$

d) $q(x) = -2x + 6$ for $q(x) = -3$

We will now find the values of x and f(x) on the graph of a function.

Practice 5

Use the graphs below to determine the indicated values.

Upside down v-shaped graph. One line is the equation y=3x+2 and the second line is y=-x+2 and they meet at the y-intercept of 2. The graph stops at x=1.5 and is continuous to the left. The y-axis is labelled with f(x).

a) Evaluate.

i) $f(1)$

ii) $f(0)$

iii) $f\left(\frac{3}{2}\right)$

b) Find the value of x.

i) $f(x)=-1$

ii) $f(x)=2$

iii) $f(x)=-1.75$

c) Evaluate.

i) $f(0)$

ii) $f(-2)$

iii) $f(1)$

d) Find the value of x.

i) $f(x)=-3$

ii) $f(x)=1$

Homework

Evaluate each of the expressions given the following functions.

a) $f(x)=2x+7$ for $f(-7)$

b) $g(x)=-3x-4$ for $g(-1.5)$

c) $h(x)=5x-9$ for $h\left(\frac{3}{5})\right$

d) $p(x)=3x^2+2x-1$ for $p(1)$

e) $q(x)=-x^2+6x+4$ for $q(-2)$

f) $r(x)=4x^2-7x+3$ for $r\left(\frac{1}{2})\right$
Consider the functions below. Evaluate for the given values.
$f(x) = x^2 + 4$
$g(x) = -2x + 5$
$h(x) = 4x - 7$

a) $f(4)$

b) $f(-3) + g(6)$

c) $h(-2) + f(7)$

d) $f(0) - g(-8)$

e) $f(-5) + g(2) - h(3)$

f) $h(-6) - g(1)$
Consider the functions below. Evaluate for the given values.
$j(x) = x^2 + 5x - 2$
$k(x) = -4x + 3$
$m(x) = 2x^2 - 6x + 1$

a) $j(3)$

b) $j(1) - k(-3)$

c) $j(-2) + k(5)$

d) $j(-4) + k(2) - m(3)$

e) $m(4) + j(0)$

f) $m(-1) - k(0)$
Given the functions below, write a simplified expression for each of the following expressions.
$F(x) = 3x - 1$
$G(x) = x^2 - 4x + 6$
$H(x) = -4x + 3$

a) $F(x - 1)$

b) $G(x + 1)$

c) $H(x + 2)$

d) $F(x) + G(x)$

e) $F(x + 2)$

f) $G(x - 1)$

g) $H(x + 3)$

h) $F(2x)$

i) $G(x + 1) + H(x)$

j) $F(x - 4) - H(x + 2)$
Solve for x. Round to the nearest hundredth, if necessary.

a) $f(x) = 3x - 5$ for $f(x) = 10$

b) $g(x) = 2x + 7$ for $g(x) = 15$

c) $h(x) = 5x^2 - 3$ for $h(x) = 17$

d) $p(x) = -x + 4$ for $p(x) = -2$

e) $q(x) = x^2 + 4$ for $q(x) = 9$

f) $M(x) = -4x + 1$ for $M(x) = 5$

g) $J(x) = 3x + 1$ for $J(x) = 7$

h) $k(x) = 7x + 2$ for $k(x) = 13$

i) $T(x) = x^2 - 1$ for $T(x) = 4$

j) $L(x) = 4x + 9$ for $L(x) = 21$
Use each graph, f(x), to determine the indicated values.

a) Evaluate.

i) $f(2)$

ii) $f(4)$

iii) $f(0)$
b) Find the value of x.

i) $f(x)=4$

ii) $f(x)=5$

iii) $f(x)=0$

c) Evaluate.

i) $f(-3)$

ii) $f(0)$

iii) $f(1)$
d) Find the value of x.

i) $f(x)=0$

ii) $f(x)=-5$

iii) $f(x)=-9$

e) Evaluate.

i) $f(-1)$

ii) $f(0)$

iii) $f(1)$
f) Find the value of x.

i) $f(x)=7$

ii) $f(x)=8$

iii) $f(x)=9$
If the input of the function $f(x)=4x-3$ is $\frac{3}{4}$ , what is the corresponding output?
If the output of the function f $f(x)=-11x+4$ is $7$ , what is the corresponding output written as an exact value?

Answers

1.

a) –7

b) 0.5

c) –6

d) 4

e) –12

f) 0.5

2.

a) 20

b) 6

c) 38

d) –17

e) 25

f) –34

3.

a) 22

b) –11

c) –25

d) –12

e) 7

f) 6

4.

a) $3x - 4$	b) $x^2 - 2x + 3$	c) $-4x - 5$	d) $x^2 - x + 5$	e) $3x + 5$
f) $x^2 - 6x + 11$	g) $-4x - 9$	h) $6x - 1$	i) $x^2 - 6x + 6$	j) $7x - 8$

5.

a) 5	b) 4	c) $\pm2$	d) 6	e) $\pm\sqrt{5}$
f) –1	g) 2	h) 1.57	i) $\pm\sqrt{5}$	j) 3

6.

a) i) 2	a) ii) 1	a) iii) 3	b) i) –2	b) ii) –4	b) iii) 6
c) i) 0	c) ii) –9	c) iii) –8	d) i) $\pm3$	d) ii) $\pm2$	d) iii) 0
e) i) 7	e) ii) 8	e) iii) 9	f) i) –1	f) ii) 0	f) iii) 1

7. 0	8. $-\frac{3}{11}$

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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