Aim: Function Operations Course: Alg. 2 & Trig. Aim: What are some ways that functions can operate?...

Aim: Function Operations

Course: Alg. 2 & Trig.

Aim: What are some ways that functions can operate?

Do Now:

Evaluate f(x) = 3x and g(x) = x – 5 if x = 3.

f(3) = 3x g(3) = x – 5f(3) = 3(3) g(3) = (3) – 5

f(3) = 9 g(3) = -2

What is the value of f(3) + g(3)? 9 + -2 = 7

What is the sum of f(x) + g(x)? 4x – 5

What is the domain for f(x) + g(x)?

All real numbers



Operation Resulting Function Domain

Function Operations

Addition f(x) + g(x) All reals

f(x) = 3x and g(x) = x – 5

= 3x + (x – 5) = 4x – 5

Subtraction f(x) – g(x) All reals

= 3x – (x – 5) = 2x + 5

Multiplication f(x) • g(x) All reals

= (3x)(x – 5) = 3x2 – 15x

Division f(x) g(x) All reals except 5

3x

x 5



h(x) 3

x 4

f(x) = -3x2 + 4x + 5

g(t ) t

Finding the Domain of a Function

All reals

All non-negative reals

What is the domain for the following functions?

What numeric inputs make valid statements?

All reals except four

Understanding domain will clarifygraphing of function.



Composition of Functions

A binary operation that processes the resultsof one function through a second function.

f(x) = x2 and g(x) = x + 4

1st input

x

f(?) = ?2

x21st

outputx2

2nd input

x2

g(?) = ? + 4

x2 + 42nd

outputx2 + 4



f(?) = ?2

Composition of Functions

f(x) = x2 and g(x) = x + 4

1st input

x42

1st output

16

2nd input

16

g(?) = ? + 4

16 + 42nd

output20

let x = 4

4 Composition



Composition of Functions Notation

f(x) = x2 and g(x) = x + 4

let x = 4

f(x)g( )

How do we symbolically show this composition of functions?

1. Evaluate the innerfunction f(x) first.

1 2. Then use the answeras the input for theouter function g(x).

2

g(f(4)) = 20



Composition of Functions Notation

f(x) 1. Evaluate the innerfunction f(x) first.

g( )

2. Then use the answeras the input for theouter function g(x).

1

2

A composite function is a new function created by the output of one function used as the input of a second function.

g f g( f (x))

g o f g of f of x

Read as “ g of f of x”



Model Problems

Find the composition of f(x) = 4x with g(x) = 2 – x

f(g(x)) = f(2 – x)

g(x)

Substitute 2 - x for g(x)

4(2 – x) Apply f( ) = 4( )8 – 4x Simplify

Find the composition of g(x) = 2 – x with f(x) = 4x

g(f(x)) = g(4x)

f(x)

Substitute 4x for f(x)

2 – (4x) Apply g( ) = 2 – ( )

2 – 4x Simplify

Composition of functions are not commutativef(g(x)) g(f(x))

note wording



Model Problems

Let h(x) = x2 and r(x) = x + 3

a. Evaluate (h r)(5)

b. Find the rule of the function (h r)(x)

a. r(x) = x + 3 r(5) = 5 + 3 = 8

h(x) = x2 h(8) = 82 = 64

b. (h r)(x) = h( r(x))

h( x + 3) h(x) = x2

h( x + 3) = (x + 3)2 = x2 + 6x + 9

(h r)(x) = x2 + 6x + 9

Evaluate x2 + 6x + 9 when x = 5



Model Problems

A store offers a 10% discount sale on its $25jeans. You also have a coupon worth $5 offany item. Which is a better deal: 10% off first then subtract the $5 or subtract the $5 and then discount the rest at 10%?

Use functional notation/operations; let x represent the original price

f(x) = x – 5 g(x) = .90xsubtract $5 10% discount

g(f(x))

subtract $5; discount 10% discount 10%; subtract $5

f(g(x))

g(x – 5) = .90(x – 5) f(.90x) = .90x – 5

g(f(25)) f(g(25))= 18 = 17.50



Hand-in Assignment

The regular price of a certain new car is $15,800.The dealership advertised a factory rebate of $1500 and a 12% discount. Compare the sale price obtained by subtracting the rebate first,then taking the discount, with the sale priceobtained by taking the discount first, thensubtracting the rebate.Use function notation as in a similar problem inyour text book (Book B) on page 28 to explainyour resulting conclusions on looseleaf – to be handed in.

Aim: Function Operations Course: Alg. 2 & Trig. Aim: What are some ways that functions can operate?...

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