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2.02 Function notation

Worksheet
Function notation
1

The point \left(7, - 6 \right) satisfies the function f(x). Write this using function notation.

2

If f \left(x\right) = 9 x^{2} + 7 x - 4, find:

a

f(-4)

b

f(10)

3

If f \left(x\right) = - 6 x + 4, find:

a

f \left( 4 \right)

b

f \left( 0 \right)

4

If f(x) = 4 x + 4, find:

a

f \left( 2 \right)

b

f \left( - 5 \right)

5

If f \left( x \right) = 3 x - 1, find:

a

f \left( 3 \right)

b

f \left( - 4 \right)

6

If g \left( x \right) = \dfrac{7 x}{4}, find:

a

g \left( 5 \right)

b

g \left( - 4 \right)

7

Consider the function f \left( x \right) = 4 + x^{3}.

a

Evaluate f \left( 4 \right)

b

Evaluate f \left( - 2 \right)

8

Consider the function f \left( x \right) = 2 x^{2} - 2 x + 5. Evaluate f\left(\dfrac{1}{2}\right).

9

Consider the function f \left( x \right) = \sqrt{ 5 x + 9}. Find the exact value of:

a

f \left( 0 \right)

b

f \left( 2 \right)

c

f \left( - 1 \right)

10

If f \left( t \right) = \dfrac{t^{3} + 27}{t^{2} + 9} , evaluate the following:

a

f \left( - 3 \right)

b

f \left( 3 \right)

c

f \left( 4 \right)

11

Consider the function f \left( x \right) = x^{2} + 8 x. Write an expression for:

a

f \left( a \right)

b

f \left( b \right)

12

Consider the function f \left( x \right) = 2 x^{3} + 3 x^{2} - 4.

a

Evaluate f \left( 0 \right).

b

Evaluate f \left( \dfrac{1}{4} \right).

13

If j(x) = 3^{x} - 3^{ - x }, find the following, rounding your answers to two decimal places if necessary:

a

j(0)

b

j(1)

c

j(4)

14

If m \left( x \right) = \sqrt{12^{2} - x^{2}}, find:

a

m \left( 4 \right)

b

m \left( 0 \right)

c

m \left( 9 \right)

d

m \left( \sqrt{2} \right)

15

Consider the function p \left( x \right) = x^{2} + 8.

a

Evaluate p \left( 2 \right).

b

Form an expression for p \left( m \right).

16

Consider the equation x + 3 y = 6.

a
Make y the subject of the equation.
b

Rewrite the equation using function notation f \left( x \right) for y.

c

Find the value of f \left( 3 \right).

17

Consider the equation x - 4 y = 8.

a
Make y the subject of the equation.
b

Rewrite the equation using function notation f \left( x \right).

c

Find the value of f \left( 12 \right).

18

Consider the equation y + 6 x^{2} = 3 - x.

a
Make y the subject of the equation.
b

Rewrite the equation using function notation f \left( x \right).

c

Find the value of f \left( 3 \right).

19

Consider the equation y - 4 x^{2} = 5 + x.

a
Make y the subject of the equation.
b

Rewrite the equation using function notation f \left( x \right).

c

Find the value of f \left( 2 \right).

20

Consider the equation - 6 x + 5 y = 7.

a
Make y the subject of the equation.
b

Rewrite the equation using function notation f \left( x \right).

c

Find the value of f \left( 3 \right).

21

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
-4
-3
-2
-1
1
2
3
4
y
22

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

a

For any input value of x, state the maximum number of distinct output values h \left(x\right) can produce.

b

Is h \left(x\right) a function? Explain your answer.

23

For each of the following pairs of variables, determine which is the independent variable and which is the dependent variable:

a

Cost of pizza and size of pizza.

b

Mark achieved on a test and time spent studying.

c

Duration of a loan and amound borrowed.

d

Time spent exercising and physical fitness level.

24

Consider the function f(x) = x^{2} - 49.

a

Find:

i

f(1)

ii

f(8)

iii

f(0)

b

If f(x) = 12, what are the possible values for x, rounded to two decimal places?

25

Consider the function g \left( x \right) = a x^{3} - 3 x + 5.

a

Form an expression for g \left( k \right).

b

Form an expression for g \left( - k \right).

c

Is g \left( k \right) = g \left( - k \right)?

d

Is g \left( k \right) = - g \left( - k \right)?

Algebra of functions
26

A function is defined as f \left( x \right) = 2 + 3 x - x^{3}. Evaluate f \left( - 2 \right) + f \left( 4 \right).

27

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

a

A \left( 5 \right)

b

Q \left( 4 \right)

c

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

28

Consider the function f \left( x \right) = x^{3} - 8 x + 7.

a

Evaluate f \left( 4 \right).

b

Evaluate f \left( - 4 \right).

c

Form an expression for f \left( a \right).

d

Form an expression for f \left( b \right).

e

Form an expression for f \left( a + b \right).

f

Is f \left( a \right) + f \left( b \right) = f \left( a + b \right) for all values of a and b?

29

Consider the function f \left( x \right) = x^{2} + 5 x. Find f(a + h).

30

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

31

Consider the function f \left( x \right) = x^{2} - 3 x - 2.

a

Form an expression for f \left( a - 2 \right).

b

Form an expression for f \left( a + h \right) - f \left( a \right).

32

Consider the function f \left( x \right) = 4 x^{2} + 4 x + 3.

a

Form an expression for f \left( a + 2 \right).

b

Form an expression for f \left( a + h \right) - f \left( a \right).

33

Consider the following table of values:

a

Find (f + g)(2).

b

Find (f + g)(7).

c

Find (f \times g)(2).

d

Find (g-f)(9).

x2789
f(x)4141618
g(x)8283236
34

Let f \left( x \right) = x^{2}-1 and g \left( x \right) = 5 x-1. Find f(2) + g(2).

35

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).

36

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)

Composition functions
37

If f \left( x \right) = 4 x^{3} and g \left( x \right) = x + 5, find:

a
\left( f \circ g\right)(x)
b
\left( f \circ g\right)(-2)
c
\left( f \circ g\right)(0)
d
\left( g \circ f\right)(x)
e
\left( g \circ f\right)(-2)
f
\left( g \circ f \right)(0)
38

Consider the functions f \left( x \right) = - 2 x - 6 and g \left( x \right) = 5 x - 7.

a

Find f \left( 7 \right).

b

Hence, or otherwise, evaluate g \left( f \left( 7 \right) \right).

c

Find g \left( 7 \right).

d

Hence, evaluate f \left( g \left( 7 \right) \right).

e

Is it true that f \left( g \left(x\right)\right) = g \left( f \left(x\right)\right) for all x?

39

Consider the functions f \left( x \right) = - 2 x - 8 and g \left( x \right) = 4 x^{2} - 4.

a

Evaluate (g \circ f )\left( 6 \right) .

b

The function h \left( x \right)= (f \circ g ) \left( x \right). Write h \left( x \right) in terms of x.

c

Are (f \circ g )\left(x\right) and (g \circ f ) \left(x\right) both quadratic functions?

40

Consider the functions f \left(x\right) = - 2 x + 2, g \left(x\right) = 4 x^{2} - 8 and r \left(x\right) = - 3 x - 8.

a

Evaluate (f \circ g) \left(6\right).

b

Evaluate (r \circ f \circ g) \left(6\right).

41

Consider f \left(x\right) = x^{2} + 3 and g \left(x\right) = 4 x - 9.

a

Define f \left( 2 x\right).

b

Show that f \left( 2 x\right) = g \left( f \left(x\right)\right).

Applications
42

If Z(y) = y^{2} + 12 y + 32, find y when Z(y) = - 3.

43

A function f \left( x \right) is defined by f \left( x \right) = \left(x + 4\right) \left(x^{2} - 4\right).

a

Evaluate f \left( 6 \right).

b

Find all solutions for which f \left( x \right) = 0.

44

A graph of a quadratic equation of the form y = a x^{2} + b x + c passes through the points \left(0, - 7 \right), \left( - 1 , - 8 \right) and \left(4, 77\right) .

a

Using the point \left(0, - 7 \right), find the value of c.

b

Substitute c and \left( - 1 , - 8 \right) into the equation y = a x^{2} + b x + c to obtain an equation that describes a in terms of b.

c

Substitute c and \left(4, 77\right) into the equation y = a x^{2} + b x + c to obtain a simplified second equation in terms of a and b.

d

Solve for a and b.

45

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

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

46

Elizabeth wants to calculate the cost to travel to Tehran and then Mexico City at certain times of the year. A program on her computer calculates T \left( x \right), the total cost of this trip, by adding the cost of travelling to Tehran, S \left( x \right), and the cost of travelling to Mexico City, U \left( x \right).

Based on historical data, the computer program uses the calculations:

S \left( x \right) = x^{2} - 200 x + 10\,227 \quad \text{ and } \quad U \left( x \right) = 18 x + 15\,423 where x is the number of days until the date of travel.

a

Find an equation for T \left( x \right).

b

Find the values of the following:

i

S \left( 91 \right)

ii

U \left( 91 \right)

iii

T \left( 91 \right)

c

Sketch the graphs of y = S \left( x \right), y = U \left( x \right) and y = T \left( x \right).

47

A conical container is being filled with water, as shown in the diagram. The water is being poured in such a way that the radius of the water's surface increases at a rate of 7\text{ cm/s}.

a

Find a function r \left( t \right) for the radius after t seconds.

b

Find a function A \left( r \right) for the area of the water's surface in terms of the radius r.

c

Find the composite function \left(A\ \circ\ r\right)(t).

d

Interpret the function \left(A\ \circ\ r\right)(t) in context.

48

Air is being added to a spherical balloon. At a time t (in seconds), the radius r of the balloon (in \text{cm}) can be given by the function r \left( t \right) = 3 \sqrt{t}. The volume of a sphere in terms of its radius is given by the formula V \left( r \right) = \dfrac{4}{3} \pi r^{3}.

a

Find the composite function (V \circ r ) \left( t \right).

b

Interpret the function \left(V \circ r \right) \left(t\right) in context.

c

Calculate the volume of the balloon after 6 seconds. Give your answer correct to two decimal places.

49

During a sale at a certain clothing store, all shirts are on sale for \$10 less than 75\% of the original price, x.

a

Write a function g that finds 75\% of x.

b

Write a function f that finds 10 less than x.

c

Construct the composite function \left( f \circ g \right) \left( x \right) =f \left( g \left( x \right) \right).

d

Hence, calculate the sale price of a shirt that has an original price of \$86.

e

If the discounts were applied in the other order, using \left(g\circ f\right)\left(x\right), would the sale price of the same shirt increase or decrease?

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Outcomes

ACMMM023

use function notation, domain and range, independent and dependent variables

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