MHF4U1: Unit#9 * Lesson #5 Name

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MHF4U1: Unit#9 – Lesson #5 Name:

9.5 Composition of Functions

Lesson:

1.

A sales person earns $ 5 per sale plus $15 for each day of work. 30% is then deducted for taxes. Complete the table showing net pay for one day of work.

Number of sales n

0

1

2

Earnings

M = 5n + 15

Net pay

P = 0.7 M

We may represent the data with mapping diagrams

3

4

In the first step, n is the input and M(n) is the output. i.e. n is the domain of M(n) and represents the range.

In the second step, M(n) is the input and P(M(n)) is the output. i.e. M(n) is the domain of P.

We can determine an equation of for P(n) (i.e. P as a function of n) by substituting.

Now,

P

0 .

7

M

and

M

5

n

15 so P(M(n))

3

n

0

.

6

9

( 5

n

15 )

Substituting one function into another is call COMPOSITION.

2.

You can think of a function as an input-output machine, where the domain is the set of input and the range is the set of output.

Input-output diagram

Input

f

Output

x f(x)

domain of f range of f

However, we can extent the input-output diagram so that the input set goes in a function and the output set from

g goes into function f, producing the function . See Diagram for illustration.

Input-output diagram

Input

x

domain of g

f g(x) g

Range of g domain of f

Output

f(g(x))

range of f

Combining two functions in this way is called Composition-substituting the output of one function into another function.

The new function is called the composite of

f

and

g

and can also be written as

(

f

.

6.

MHF4U1: Unit#9 – Lesson #5

3.

Name:

If 𝑓(𝑥) = 2𝑥 a) ) f (1)

2

and 𝑔(𝑥) = −3𝑥 + 1 . Determine the following b) g (x

3) c) 𝑓(𝑔(𝑥))

4.

5.

d)

(

f

g

)( 1 )

Given that a)

f

g f

(

x

)

2

,

x g

(

x

)

x

1

, find expressions for following and state their domains and ranges. b)

g

f

For each function h(x), find functions f and g such that

h

 a) h(x) = (x

3

+ 2)

3 b)

h

(x) =

x

3 c)

f

1 

f f

g

. c)

h

(x) = log(

x

2

2 ) d)

g

g

d)

h (x) = 4x

4

93

Given

f

(

x

)

2

x

1

and g

(

x

)

 cos

x

, describe

f g

as a transformation of the graph

y

.

7.

A pebble dropped into a pond creates a ripple that propagates outward in all directions at a constant rate of 5 m/min.

a)

Express the radius, r, in metres, of the circular area enclosed by the ripple as a function of the time, t, in

b)

If A is the circular area, in square metres, as a function of the radius, find

A

r

and explain what it minutes. represents.

c) What is the circular area 10 min after the pebble is dropped into the pond?

Homework: Pg. 552 #1-3, 5-7, 9-11, 14.

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