Lessons on Composition and Combination of Functions

Function Combination & Composition - Part 2

In the previous lesson, you were introduced to combining functions using algebraic operations.

Function Combination & Composition

Fancy ways of combining like terms, multiplying and dividing polynomials, and performing substitution.

Objectives: SWBAT

• Review combining functions using algebraic operations.

• Evaluate composite functions.

• Create a new function by composition of functions.

Multiple Choice

$$f\left(x\right)$$  is the same as ____.

• y

• input

• x

• an equation

Multiple Choice

The function $$h\left(t\right)$$   represents the height in inches of a plant after $$t$$  weeks. What does the notation  $$h\left(4\right)\ =\ 5$$  mean?

• After 5 weeks the plant is 4 inches tall.

• The plant grows at a rate of (5/4) inches per week.
  
  

• After 4 weeks the plant is 5 inches tall.

• The plant grows at a rate of 1 inch per week.

Open Ended

What ordered pair represents $$h\left(4\right)\ =\ 5$$ ? 

Anatomy of Function Notation

h(t) = 5 ------------> h(4) = 5

t is the input

h(t) is the output

(input, output)

(t, h(t))

(4, 5)

Combining Functions

Multiple Choice

$$f\left(x\right)=2x^2+3x-5\ \ and\ \ -x^2+5x\ +1.\$$  Find  $$\left(f+g\right)\left(x\right)\ \ and\ \ \left(f-g\right)\left(x\right).$$  

• $$x^2+8x+4\ and\ 3x^2+8x-4$$  

• $$x^2+8x-4\ and\ 3x^2-2x-6$$  

• $$x^2+8x-4\ and\ 3x^2-2x-6$$  

Poll

Check for Understanding.

• I understand.

• I need more resources.

• I need to attend Instructional Support sessions.

Evaluating Functions

Fancy word for SUBSTITUTION!

Multiple Choice

For the function  $$f\left(2\right)=3x-1$$  what is the input and output represented as an ordered pair?

• $$\left(5,2\right)$$  

• $$\left(2,5\right)$$  

• $$\left(5,3\right)$$  

• $$\left(3,5\right)$$  

Multiple Choice

Evaluate  y = 4x - 1 when x = 10.

• $$38$$

• $$39$$  

• $$40$$  
  

• $$41$$  
  

Fill in the Blank

Evaluate  $$f\left(10\right)=4x-1.$$  

Fill in the Blank

Given  $$f\left(x\right)=-8x⁴$$   and

$$g\left(x\right)=5x³.$$  Find  $$\left(fg\right)\left(1\right).$$  

Back to the DO NOW!

On November 25th, a store selling very expensive clothing will sell any item for $50 less than the listed price. On any day in November, the store will give a discount of 15% to any customer who can prove that he/she contributed to a local charity. You go to the store on November 25th with evidence you have donated to a local charity. Is it better for you to subtract $50 and then apply 15% discount or apply 15% discount and then subtract $50? Explain.

Multiple Choice

What was your choice of order?

• subtract $50 and then apply 15% discount

• Apply 15% discount and then subtract $50

• Not sure

Open Ended

What is the independent variable in the scenario?

Multiple Select

What other terms can you use to describe independent variable? Check all that apply.

• domain

• input

• output

• range

• x-values

Multiple Choice

If x represents the listed price of an item in the store, what function can represent the price you will pay on November 25th?

• $$p\left(x\right)=x-50$$  

• $$p\left(x\right)=0.85x$$  

• $$y=x-50$$  

• $$y=0.85x$$  

Multiple Choice

How can you represent the price discounted at 15% in the month of November, using function notation?

• $$y=0.15\%$$  

• $$y=0.85\%$$  

• $$d\left(x\right)=0.15x$$  

• $$d\left(x\right)=0.85x$$  

• $$d\left(x\right)=0.85x\%$$  

Open Ended

Reflection. Looking back at the DO NOW! exercise, have

you changed your mind? Explain.

• Fancy term for substitution. (Performed at least twice)

• A composite function is created when one function is substituted into another function.

• A composite function consists of applying one rule, getting a result, and then applying a second rule to the result obtained from the first rule.

• Decisions are made in which the order or sequence of application can and will make a difference.

The key idea in function composition is the input of the function is NOT a numerical value, instead the INPUT is ANOTHER FUNCTION.

When the output of one function is used as the input of another, the entire operation is called a composition of functions.

Model Problem

f(x) = 5x + 3

g(x) = 3x² . Find f(g(4)).

• Step 1: Substitute 4 for x in g(x)

• g(4) = 3(4)²

• g(4) = 3(16)

• g(4) = 48

• Step 2: Substitute 48 into f(x).

• f(48) = 5(48) + 3

• f(48) = 240 + 3 = 243 ... f(g(4)) = 243

Multiple Choice

$$f\left(x\right)=5x+3$$  

$$g\left(x\right)=3x²$$  

$$Find\ g\left(f\left(4\right)\right).$$  

• 243

• 1104

• 1587

• None of these

Multiple Select

Is  $$f\left(g\left(x\right)\right)$$  the same as  $$g\left(f\left(x\right)\right)$$  ?

• Yes

• No

Poll

Check for Understanding.

• I understand.

• I need more practice.

• I need to attend Instructional Support.