Grade 12 Pre-Calculus Mathematics

[MPC40S]

Chapter 10

Function Operations

Outcome

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12P.R.1. Demonstrate an understanding of operations on, and compositions of, functions.

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Chapter 10 – Homework

Section Page Questions

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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:______________________

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Example 2

a) Given the graphs of 𝑓(𝑥) = 𝑥 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

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Example 3 Given the graphs of 𝑓(𝑥) and 𝑔(𝑥). Sketch the graph of (𝑔 − 𝑓)(𝑥) Domain:_________________________________

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Example 4 Use the following graphs to sketch the graph of 𝑔(𝑥). )(xf xgf

𝑥 𝑓(𝑥) 𝑔(𝑥) (𝑓 − 𝑔)(𝑥)

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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 ∙ 𝑔)(𝑥) (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 ( )

( )

Domain:__________________________

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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

(𝑥)

c) Evaluate the following expressions

a) (1) b) (𝑓 ⋅ 𝑔)(−3) c) (0)

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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) = 𝑥 + 1 and 𝑔(𝑥) = 2𝑥 − 3

a) Determine the equation for 𝑓(𝑔(𝑥))

b) Determine the equation for (𝑔 ∘ 𝑓)(𝑥)

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c) Determine the equation for 𝑔(𝑔(𝑥)) d) Evaluate 𝑓(𝑔(2))

e) Evaluate (𝑔 ∘ 𝑓)(1)

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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)

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Example 3 Given 𝑓(𝑥) = 2𝑥 − 3 and 𝑔(𝑥) = , 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.

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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.

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Example 5 Given 𝑓(𝑥) = √𝑥 + 3 and 𝑔(𝑥) = 𝑥 − 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.

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d) Graph 𝑔 𝑓(𝑥) .

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CHAPTER 10 REVIEW: Old Exam Questions Outcome R1 June 2015

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January 2015

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June 2014

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January 2014

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June 2013

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January 2013

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