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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 = *5*n *+ 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.