Upload
others
View
10
Download
0
Embed Size (px)
Citation preview
Grade 12 Pre-Calculus Mathematics
[MPC40S]
Chapter 10
Function Operations
Outcome
R1
12P.R.1. Demonstrate an understanding of operations on, and compositions of, functions.
MPC40S Date:______________
2
MPC40S Date:______________
3
Chapter 10 – Homework
Section Page Questions
MPC40S Date:______________
4
Chapter 10: FUNCTION OPERATIONS
10.1 – Sums and Differences of Functions
We can form new functions by performing operations with functions. To combine two functions, 𝑓(𝑥) and 𝑔(𝑥), add or subtract as follows: Sum of Functions Difference of Functions 𝑓(𝑥) + 𝑔(𝑥) 𝑓(𝑥) − 𝑔(𝑥) (𝑓 + 𝑔)(𝑥) (𝑓 − 𝑔)(𝑥) Example 1
Given 𝑓(𝑥) = 𝑥 + 1 and 𝑔(𝑥) = 2𝑥 − 3. a) Write an equation to represent 𝑓(𝑥) + 𝑔(𝑥) Domain:____________________ b) Write an equation to represent 𝑓(𝑥) − 𝑔(𝑥)
Domain:______________________
R1
MPC40S Date:______________
5
Example 2
a) Given the graphs of 𝑓(𝑥) = 𝑥2 and 𝑔(𝑥) = −2𝑥 + 1 below sketch (𝑓 + 𝑔)(𝑥)
b) Write an equation to represent (𝑓 + 𝑔)(𝑥)
𝑥 𝑓(𝑥) 𝑔(𝑥) 𝑓(𝑥) + 𝑔(𝑥)
−2
−1
0
1
2
MPC40S Date:______________
6
Example 3
Given the graphs of 𝑓(𝑥) and 𝑔(𝑥). Sketch the graph of (𝑔 − 𝑓)(𝑥) Domain:_________________________________
MPC40S Date:______________
7
Example 4
Use the following graphs to sketch the graph of 𝑔(𝑥).
)(xf ( )( )xgf −
𝑥 𝑓(𝑥) 𝑔(𝑥) (𝑓 − 𝑔)(𝑥)
MPC40S Date:______________
8
Chapter 10: FUNCTION OPERATIONS
10.2 – Products and Quotients of Functions
We can form new function by performing operations with functions. To combine two functions, 𝑓(𝑥) and 𝑔(𝑥), multiply or divide as follows: Product of Functions Quotient of Functions
𝑓(𝑥)𝑔(𝑥) 𝑓(𝑥)
𝑔(𝑥)
𝑓 ∙ 𝑔)(𝑥) (𝑓
𝑔) (𝑥)
Example 1
Given the functions 𝑓(𝑥) = 𝑥 + 1 and 𝑔(𝑥) = 2𝑥 − 3
a) Write an equation for 𝑓(𝑥) ∙ 𝑔(𝑥) Domain:_________________________
b) Write an equation for 𝑔(𝑥)
𝑓(𝑥)
Domain:__________________________
R1
MPC40S Date:______________
9
Example 2
Using the graphs of 𝑓(𝑥) and 𝑔(𝑥) given below, answer the following questions:
a) Identify the zeros of (𝑓 ∙ 𝑔)(𝑥). b) State the vertical asymptote(s) of
(𝑓
𝑔) (𝑥)
c) Evaluate the following expressions
a) (𝑔
𝑓) (1) b) (𝑓 ⋅ 𝑔)(−3) c) (
𝑓
𝑔) (0)
MPC40S Date:______________
10
Chapter 10: FUNCTION OPERATIONS
10.3 - Composite Functions
Composite Functions:
- Are functions that are formed from two functions, 𝑓(𝑥) and 𝑔(𝑥) , in which the output (or result) of one of the functions is used as the input for the other function.
- The composition of 𝑓(𝑥) and 𝑔(𝑥) is defined as ______________ and is formed
when the equation of 𝑔(𝑥) is substituted into the equation of 𝑓(𝑥).
- ______________ is read as _____________________________.
- We can write composite function as _______________ or _______________.
- Composite functions must not be confused with multiplication. That is (𝑓 ∘ 𝑔)(𝑥) is not the same as (𝑓 ∙ 𝑔)(𝑥).
Example 1
Given 𝑓(𝑥) = 𝑥2 + 1 and 𝑔(𝑥) = 2𝑥 − 3
a) Determine the equation for 𝑓(𝑔(𝑥))
b) Determine the equation for (𝑔 ∘ 𝑓)(𝑥)
R1
MPC40S Date:______________
11
c) Determine the equation for 𝑔(𝑔(𝑥)) d) Evaluate 𝑓(𝑔(2))
e) Evaluate (𝑔 ∘ 𝑓)(1)
MPC40S Date:______________
12
Example 2
Use the table below to evaluate the following.
x 1 2 3 4 5 6
𝑓(𝑥) 3 1 4 2 2 5
𝑔(𝑥) 6 3 2 1 2 3
a) 𝑓(𝑔(2)) b) 𝑔(𝑓(0))
c) (𝑔 ∘ 𝑓)(3)
d) (𝑓 ∘ 𝑓)(1)
MPC40S Date:______________
13
Example 3
Given 𝑓(𝑥) = 2𝑥 − 3 and 𝑔(𝑥) =𝑥+3
2, determine 𝑓(𝑔(𝑥)) and 𝑔(𝑓(𝑥)).
𝑓(𝑔(𝑥)) 𝑔(𝑓(𝑥))
Recall: When 𝑓(𝑔(𝑥)) = 𝑥 or 𝑔(𝑓(𝑥)) = 𝑥, this means that 𝑓(𝑥) and 𝑔(𝑥) are
__________________ of each other.
MPC40S Date:______________
14
Example 4
Use the graphs below to answer the following questions:
a) 𝑓(𝑔(2)) b) 𝑔(𝑓(4))
c) Determine the value of x if f(g(x)) = 3.
MPC40S Date:______________
15
Example 5
Given 𝑓(𝑥) = √𝑥 + 3 and 𝑔(𝑥) = 𝑥2 − 4. a) Domain of 𝑓(𝑥) _____________ Domain of 𝑔(𝑥) ________________
b) Determine 𝑓(𝑔(𝑥)) and identify the domain.
c) Determine 𝑔(𝑓(𝑥)) and identify the domain.
MPC40S Date:______________
16
d) Graph 𝑔(𝑓(𝑥)).
MPC40S Date:______________
17
CHAPTER 10 REVIEW: Old Exam Questions Outcome R1
June 2015
MPC40S Date:______________
18
January 2015
MPC40S Date:______________
19
June 2014
MPC40S Date:______________
20
MPC40S Date:______________
21
MPC40S Date:______________
22
January 2014
MPC40S Date:______________
23
June 2013
MPC40S Date:______________
24
MPC40S Date:______________
25
MPC40S Date:______________
26
January 2013
MPC40S Date:______________
27