Objectives:
1.
To find the difference quotient of a function
2.
To add, subtract, multiply, and divide functions and find their domain
3.
To find the composition of two functions and find its domain
Assignment:
• P. 89: 5-12 S
• P. 90: 35-42 S
• P. 90: 47-54 S
• Homework
Supplement
• Read: P. 93-98

Difference Quotient Pascal’s Triangle
Function Composition
Function
Decomposition

You will be able to find the difference quotient of a function

Given 𝑓 𝑥 = 4𝑥 2 − 2𝑥 , find 𝑓 𝑥 + ℎ .

In calculus, the difference quotient is defined as 𝑓 𝑥 + ℎ − 𝑓 𝑥 ℎ where ℎ ≠ 0.
Eventually, you will learn to take the limit of this quotient as h approaches 0. That’s known as the derivative of the function. But for now, we’re just going to evaluate the thing.

In calculus, the difference quotient is defined as 𝑓 𝑥 + ℎ − 𝑓 𝑥 ℎ where ℎ ≠ 0.
It’s just the slope of a secant line!

Given 𝑓 𝑥 = 4𝑥 2 quotient.
− 2𝑥, find the difference

1
1 1
1 2 1
1
1
4
3 3
6 4
1
1
1.
Evaluate f ( x + h ) separately.
2.
Evaluate f ( x + h ) first, then find the difference quotient.
3.
Use Pascal’s Triangle to help evaluate
( x + h ) n .

Coefficients:
1
1 1
 x
 h

0 
1
 x
 h

1 
1
1
1
4
2
3 3
6
1
4
1
 x
 h

2  x
2 
2 xh
 h
2
 x
 h

3  x
3

3
2 x h

3 xh
2
 h
3
1
 x
 h

4  x
4 
4
3 x h

6
2 2 x h

4 xh
3  h
4

Coefficients:
1
1 1
 x
 h

0 
1
 x
 h

1 
1
1
1
4
2
3 3
6
1
4
1
 x
 h

2  x
2 
2 xh
 h
2
 x
 h

3  x
3

3
2 x h

3 xh
2
 h
3
1
 x
 h

4  x
4 
4
3 x h

6
2 2 x h

4 xh
3  h
4

Given 𝑓 𝑥 = 2𝑥 3 quotient.
+ 𝑥 2 , find the difference

You will be able to add, subtract, multiply, and divide functions and find their domain

A function is a relation in which each input has exactly one output.
• A function is a
dependent relation
• Output depends on the input
Domain:
The set of all input values
Range:
The set of all output values

Remember adding, subtracting, multiplying, and dividing numbers? Well, we can do all of those things with functions to make new functions!
When +/− functions:
When
 functions:
When
 functions:
Just collect like terms
Use the distributive property
Write functions as a quotient and simplify

Remember adding, subtracting, multiplying, and dividing numbers? Well, we can do all of those things with functions to make new functions!

Remember adding, subtracting, multiplying, and dividing numbers? Well, we can do all of those things with functions to make new functions!
Sum: ( f

)( )

( )

( )
Difference: ( f

)( )

( )

( )
Product:
Quotient:
( )( )

( )

( )
 
 

, ( )

0

The domain of h consists of all x -values that are common to domains of BOTH f and g .
Domain of f
Domain of h
Domain of g h
 f
 g
 f
 g
 f
 g
 f
 g
Except where g(x) =0

The domain of h consists of all x -values that are common to domains of BOTH f and g .
f g
Domain of h h
 f
 g
 f
 g
 f
 g
 f
 g
Except where g(x) =0

The domain of h consists of all x -values that are common to domains of BOTH f and g .
f g
Domain of h h
 f
 g
 f
 g
 f
 g
 f
 g
Except where g(x) =0

Let 𝑓 𝑥 = 2𝑥 + 1 and 𝑔 𝑥 = 𝑥 2 + 3𝑥 − 10 .
Perform the following operations and state the domain of each.
1.
f + g
2.
g + f
3.
f – g
4.
g – f

Let 𝑓 𝑥 = 𝑥 2 − 5𝑥 and 𝑔 𝑥 = 2𝑥 − 3 .
Perform the following operations and state the domain of each.
1.
𝑓 ∙ 𝑔
2.
𝑔 ∙ 𝑓

Let 𝑓 𝑥 = 𝑥 and 𝑔 𝑥 = 9 − 𝑥 2 . Perform the following operations and state the domain of each.
1.
𝑓 𝑔
2.
𝑔 𝑓

Just like with real numbers, function addition and multiplication are commutative . In other words, the order doesn’t matter.
( )

( )

( )

( )
( )

( )

( )

( )
I wonder if other properties hold as well…

A small company sells computer printers over the Internet. The company’s total monthly revenue ( R ) and costs ( C ) are modeled by the functions R ( x ) = 120 x and C ( x ) = 2500 + 75 x , where x is the number of printers sold.
1.
Find R ( x ) – C ( x )
2.
Explain what this difference means

You will be able to find the composition of two functions and find its domain

Function composition happens when we take a whole function and substitute it in for x in another function.
( )
 
( )

Substitute f(x) in for x in g(x)
– The “interior” function gets substituted in for x in the “exterior” function

Function composition happens when we take a whole function and substitute it in for x in another function.
( )
  g f

( )
Substitute f(x) in for x in g(x)
– Read as “ g composed with f ”

In function composition, the domain of h is the set of x in the domain of f such that f ( x ) is in the domain of g .
( )
 
( )

Domain of 𝑓 Range of 𝑓 𝒙 𝒇(𝒙) 𝒈 𝒇 𝒙
Domain of 𝑔 Range of 𝑔

In other words, we can only use x values that are in the domain of the interior function whose output is also in the domain of the exterior function.
( )
 
( )

Domain of 𝑓 Range of 𝑓 𝒙 𝒇(𝒙)
Domain of 𝑔 𝒈 𝒇 𝒙
Range of 𝑔

In other other words, the domain of h is the domain of the interior function f as long as its range is completely within the domain of the exterior function g .
( )
 
( )

Domain of 𝑓 Range of 𝑓 𝒙 𝒇(𝒙) 𝒈 𝒇 𝒙
Domain of 𝑔 Range of 𝑔

Let 𝑓(𝑥) = 3𝑥– 14 and 𝑔(𝑥) = 𝑥 2 𝑔 𝑓 4 .
+ 5 . Find
To do this:
1.
Find 𝑓(4) = 𝑘 .
2.
Then find 𝑔(𝑘) .

Let 𝑓(𝑥) = 𝑥 and 𝑔(𝑥) = 𝑥 3 – 2 . Find the following compositions and then state the domain.
1.
𝑓 𝑔 𝑥
2.
𝑔 𝑓 𝑥

Let 𝑓(𝑥) = 𝑥 and 𝑔(𝑥) = −𝑥 2 . Find the following composition and then state the domain.
1.
𝑔 𝑓 𝑥

Let 𝑓(𝑥) = 𝑥 2 – 9 and 𝑔(𝑥) = 9 − 𝑥 2 . Find the following composition and then state the domain.
1.
𝑓 ∘ 𝑔 𝑥

Let 𝑓(𝑥) = 3𝑥 − 8 and 𝑔(𝑥) = 2𝑥 2 . Find the following compositions.
1.
𝑔 ∘ 𝑓 5
2.
𝑓 ∘ 𝑔 5
3.
𝑓 𝑓 5
4.
𝑔 𝑔 5

Let 𝑓(𝑥) = 2 and 𝑔(𝑥) = 2𝑥 + 7 . Find the 𝑥 following compositions and state the domain of each.
1.
𝑔 𝑓 𝑥
2.
𝑓 ∘ 𝑔 𝑥
3.
𝑓 𝑓 𝑥

Rather than composing two functions to make one function, sometimes we want to take a single function h ( x ) and find two functions f ( x ) and g ( x ) whose composition yields h ( x ). This is called decomposition .
( )

2 x

3
( )

(2 x

3)
2
( )
 x 2
( )
  f g

( )

Write the function below as a composition of two functions.

3 8
 x
5

Given the functions below, find 𝑓 ∘ 𝑔 ∘ ℎ 𝑥 .
( )

2 x

3
( )
 x
2

3 x

1

Use the graphs of f and
g to graph the following:
1.
𝑓 + 𝑔
2.
𝑓– 𝑔
3.
𝑓 ‧ 𝑔
4.
𝑓 ∘ 𝑔

Objectives:
1.
To find the difference quotient of a function
2.
To add, subtract, multiply, and divide functions and find their domain
3.
To find the composition of two functions and find its domain
Assignment:
• P. 89: 5-12 S
• P. 90: 35-42 S
• P. 90: 47-54 S
• Homework
Supplement
• Read: P. 93-98