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2.02 Evaluation and operations on functions

Worksheet
Evaluate functions
1

For the following functions, determine the output produced by the indicated input value:

a

f \left(x\right) = - 6 x + 4; x = 4

b

f \left(x\right) = 9 x^{2} + 7 x - 4; x = - 4

c

g \left(t\right) = - \dfrac{2}{t} + 8; t = - 8

d

h \left(t\right) = 9^{t} + 6; t = 2

2

Rewrite the following statements using function notation:

a

Suppose that \left(7, - 6 \right) is an ordered pair that satisfies the function g.

b

In the equation x - 4 y = 8, y is a function of x.

3

Use the graph of the function f \left( x \right) to find each of the following values:

a

f \left( 0 \right)

b

f \left(- 2 \right)

c

The value of x such that f \left( x \right) = 3.

-4
-3
-2
-1
1
2
3
4
x
-3
-2
-1
1
2
3
4
5
6
7
y
4

Consider the graph of y = f \left( x \right):

a

Determine the function value at x = 3.

b

Is the graph continuous at x = 3?

-2
-1
1
2
3
4
5
6
7
8
x
1
2
3
4
5
6
7
8
9
10
y
5

Consider the function h \left(x\right) = - x^{2} + 6 x - 6.

a

For each input value of x, state the maximum number of distinct output values h(x) can produce.

b

Is h \left(x\right) a function?

6

For each function, find the function value as indicated. Round your answers to two decimal places where necessary.

a
f \left( x \right) = 3 x - 1; f \left( 3 \right)
b
f \left( x \right) = 3 x - 1; f \left( - 4 \right)
c
f \left( x \right) = \dfrac{2 x - 3}{5}; f \left( 4 \right)
d
f \left( x \right) = \dfrac{2 x - 3}{5}; f \left( - 5 \right)
e
f \left( x \right) = 4 + x^{3}; f \left( 4 \right)
f
f \left( x \right) = 4 + x^{3}; f \left( - 2 \right)
g
f \left( x \right) = 2 x^{2} - 2 x + 5; f \left( \dfrac{1}{2} \right)
h
f \left( x \right) = \sqrt{ 5 x + 9}; f \left( 0 \right)
i
f \left( x \right) = \sqrt{ 5 x + 9}; f \left( 2 \right)
j
f \left( x \right) = \sqrt{ 5 x + 9}; f \left( - 1 \right)
k
f \left( t \right) = \dfrac{t^{3} + 27}{t^{2} + 9}; f \left( - 3 \right)
l
f \left( t \right) = \dfrac{t^{3} + 27}{t^{2} + 9}; f \left( 3 \right)
m
f \left( t \right) = \dfrac{t^{3} + 27}{t^{2} + 9}; f \left( 4 \right)
n
g \left( x \right) = \sqrt[3]{1 - x}; g \left( - 6 \right)
o
g \left( x \right) = \sqrt[3]{1 - x}; g \left( 9 \right)
p
f(x) = x^{2} - 49; f\left(1\right)
q
f(x) = x^{2} - 49; f\left(8\right)
r
f(x) = x^{2} - 49; f\left(0\right)
s
f \left( x \right) = x^{3} - 8 x + 7; f \left( 4 \right)
t
f \left( x \right) = x^{3} - 8 x + 7; f \left( - 4 \right)
u
f \left( x \right) = \left(x + 4\right) \left(x^{2} - 4\right); f \left( 6 \right)
v
x - 4 y = 8; f \left( 12 \right)
7

Consider the function f(x) = x^{2} - 49. If f(x) = 12, find the values of x. Round your answer to two decimal places.

8

Consider the function f \left( x \right) = \left(x + 4\right) \left(x^{2} - 4\right). If f \left( x \right) = 0, find the values of x.

9

Consider the function Z(y) = y^{2} + 12 y + 32. If Z(y) = - 3, find the values of y.

10

Form an expression for each of the following:

a
f \left( b \right) if f \left( x \right) = x^{2} + 8 x
b
f \left( a \right) if f \left( x \right) = x^{2} + 8 x
c
f \left( a + h \right) if f \left( x \right) = x^{2} + 5 x
d
f \left( a \right) if f \left( x \right) = x^{3} - 8 x + 7
e
f \left( b \right) if f \left( x \right) = x^{3} - 8 x + 7
f
f \left( a + b \right) if f \left( x \right) = x^{3} - 8 x + 7
g
f \left( a - 2 \right) if f \left( x \right) = x^{2} - 3 x - 2
h
f \left( a + h \right) - f \left( a \right) if f \left( x \right) = x^{2} - 3 x - 2
11

If A \left( x \right) = x^{2} + 1 and Q \left( x \right) = x^{2} + 9 x, evaluate:

a

A \left( 5 \right)

b

Q \left( 4 \right)

c

A \left( 3 \right) + Q \left( 2 \right)

12

Consider the function f \left( x \right) = x^{3} - 8 x + 7. Does f \left( a \right) + f \left( b \right) = f \left( a + b \right) for all values of a and b? Show working to justify your answer.

Operations on functions
13

Suppose f and g are functions. Find the corresponding point on the graph of y = g \left( x \right), if:

a

\left(9, 12\right) is a point on the graph of y = f \left( x \right) and g \left( x \right) = f \left( x \right) - 8.

b

\left(6, - 7 \right) is a point on the graph of y = f \left( x \right) and g \left( x \right) = f \left( x - 5 \right).

c

\left(9, - 12 \right) is a point on the graph of y = f \left( x \right) and g \left( x \right) = 6 f \left( x \right).

14

Given the table of values, find \left(f + g\right)\left(2\right).

x2789
f(x)4141816
g(x)8283632
15

Given the table of values, find (f\times g)\left(6\right).

x2569
f(x)4101218
g(x)8202436
16

Let f \left( x \right) = x^{2} + 6 and g \left( x \right) = 5 x - 3.

a

Find \left(f - g\right) \left(x\right).

b

Evaluate \left(f - g\right)\left(5\right)

17

Let f \left( x \right) = - 5 x + 3 and g \left( x \right) = x^{2} - 7.

a

Find \left(f \times g\right)\left(x\right)

b

Evaluate \left(f\times{g}\right)\left( - 3 \right)

18

If f(x) = 3 x - 5 and g(x) = 5 x + 7, find:

a

(f+g)(x)

b

(f+g)\left(4\right)

c

(f-g)(x)

d

(f-g)\left(10\right)

19

For each of the following functions, find:

i
f \left( x^{2} \right)
ii
f \left( x \right)^{2}
iii
f \left( - x \right)
a
f \left( x \right) = 5 x + 4
b
f \left( x \right) = x^{2} + 4
c
f \left( x \right) = x^{3} + 2
20

For the function f \left( x \right) = x^{2} + 2 x, find \dfrac{f \left( 6 + h \right) - f \left( 6 \right)}{h}.

21

For the function f \left( x \right) = x^{2} + 3 x, find \dfrac{f \left( x + h \right) - f \left( x \right)}{h}.

22

For the function f \left( x \right) = x^{2} + 2 x, find \dfrac{f \left( p \right) - f \left( q \right)}{p - q}.

Applications
23

The financial team at Kerzon Corp. wants to calculate the profit (in dollars), P \left( x \right), generated by producing x units of personalised stationery.

The revenue (in dollars) produced by the product is given by the equation is R \left( x \right) = - \dfrac{x^{2}}{4} + 50 x. The cost of production (in dollars) is given by the equation C \left( x \right) = 12 x + 14.

The profit is calculated as P \left( x \right) = R \left( x \right) - C \left( x \right).

a

Find an equation for P \left( x \right) in terms of x.

b

Find the values of the following:

i
R \left( 76 \right)
ii
C \left( 76 \right)
iii
P \left( 76 \right)
24

The financial team at The Gamgee Cooperative wants to calculate the profit (in dollars), P \left( x \right), generated by producing x units of wetsuits.

The revenue (in dollars) produced by the product is given by the equation is R \left( x \right) = - \dfrac{x^{2}}{4} + 40 x. The cost of production (in dollars) is given by the equation C \left( x \right) = 5 x + 410.

The profit is calculated as P \left( x \right) = R \left( x \right) - C \left( x \right).

a

Find an equation for P \left( x \right) in terms of x.

b

Find the values of the following:

i
R \left( 70 \right)
ii
C \left( 70 \right)
iii
P \left( 70 \right)
c

Sketch the graphs of y = R \left( x \right), y = C \left( x \right) and y = P \left( x \right) on the same number plane.

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Outcomes

MA11-1

uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions to problems

MA11-2

uses the concepts of functions and relations to model, analyse and solve practical problems

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