Function Junction: Mastering Composition of Functions

Algebra 2 Grades High School 4:12 Video
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Lesson Description

Explore the composition of functions, understand the notation, and learn how to evaluate and simplify composite functions. Discover domain restrictions and apply your knowledge through examples.

Video Resource

Composition of Functions (Examples)

Mario's Math Tutoring

Duration: 4:12
Watch on YouTube

Key Concepts

  • Composition of functions notation
  • Evaluating composite functions
  • Simplifying composite functions
  • Domain restrictions of composite functions

Learning Objectives

  • Students will be able to correctly use the notation for composition of functions.
  • Students will be able to evaluate composite functions for numerical values.
  • Students will be able to simplify composite functions algebraically.
  • Students will be able to determine domain restrictions for composite functions.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the basic concept of a function and its notation. Introduce the concept of composition of functions as a process where the output of one function becomes the input of another.
  • Notation for Composition of Functions (5 mins)
    Explain the notation f(g(x)) and (f ∘ g)(x). Emphasize that the inner function, g(x), is evaluated first, and its output is then used as the input for the outer function, f(x). Clarify that the 'circle' symbol does not represent multiplication.
  • Evaluating Composite Functions (15 mins)
    Work through Example 1 from the video (f(g(2))). Demonstrate the step-by-step process of first evaluating g(2) and then using that result as the input for f(x). Then work through Examples 2-5 where you compose functions with variables. Address finding the domain restrictions, as shown in the video.
  • Practice and Application (15 mins)
    Have students work independently or in pairs on similar problems to the video's examples. Provide feedback and address any misconceptions. Include problems that require simplification and domain identification.
  • Bonus Challenge (5 mins)
    Present the bonus question from the video (h(m(f(x)))) as an optional challenge for students who have mastered the basic concepts. Discuss the solution as a class.

Interactive Exercises

  • Function Machine
    Use a visual aid or online tool to represent functions as machines. Students input a value, and the machine performs the function's operation. Then, connect two machines to represent the composition of two functions, visually demonstrating the process.
  • Card Sort
    Prepare cards with different functions. Students work in groups to match cards and correctly write composite functions in the correct notation.

Discussion Questions

  • How is the composition of functions different from multiplying two functions?
  • Why is it important to consider the domain restrictions when working with composite functions?
  • Can the order of composition change the final result? Explain with an example.

Skills Developed

  • Symbolic manipulation
  • Problem-solving
  • Critical thinking
  • Analytical skills

Multiple Choice Questions

Question 1:

What does the notation (f ∘ g)(x) represent?

Correct Answer: f(g(x))

Question 2:

If f(x) = x + 2 and g(x) = x², what is f(g(3))?

Correct Answer: 11

Question 3:

Given h(x) = 1/x, what is the domain restriction for h(x)?

Correct Answer: x ≠ 0

Question 4:

If m(x) = √x, what is the domain restriction for m(x)?

Correct Answer: x ≥ 0

Question 5:

If p(x) = 2x - 1 and q(x) = x + 3, what is p(q(x))?

Correct Answer: 2x + 5

Question 6:

If r(x) = x² and s(x) = x - 4, what is r(s(x))?

Correct Answer: x² - 8x + 16

Question 7:

If f(x) = 3x + 1 and g(x) = √x, what is the domain restriction on g(f(x))?

Correct Answer: x ≥ -1/3

Question 8:

Which operation is performed first in the composition of functions?

Correct Answer: The inner function

Question 9:

If f(x) = x² and g(x) = 2x, what is (f ∘ g)(1)?

Correct Answer: 4

Question 10:

Given f(x) = x / (x - 1), what value of x is NOT in the domain of f(x)?

Correct Answer: 1

Fill in the Blank Questions

Question 1:

The notation f(g(x)) means that the function _____ is being inputted into the function _____.

Correct Answer: g(x), f(x)

Question 2:

When evaluating f(g(2)), you first evaluate _____.

Correct Answer: g(2)

Question 3:

If h(x) = 4/x, x cannot equal _____.

Correct Answer: 0

Question 4:

Domain restrictions occur when you divide by _____ or take the square root of a _____ number.

Correct Answer: zero, negative

Question 5:

If f(x) = x + 5 and g(x) = x², then f(g(x)) = _____.

Correct Answer: x² + 5

Question 6:

If m(x) = 3x - 2 and n(x) = x/2, then m(n(x)) = _____.

Correct Answer: (3x/2) - 2

Question 7:

The domain of a function is the set of all possible _____ values.

Correct Answer: input

Question 8:

The output of the inner function becomes the _____ of the outer function.

Correct Answer: input

Question 9:

If f(x) = √x and g(x) = x - 3, then the domain restriction on f(g(x)) is x ≥ _____.

Correct Answer: 3

Question 10:

When finding the domain of a composite function, you must consider the domain restrictions of both the _____ and _____ functions.

Correct Answer: inner, outer

Educational Standards

A2.F.1:Understand functions as descriptions of covariation (how related quantities vary together). A2.F.1.1:Use algebraic, interval, and set notations to specify the domain and range of various types of functions, and evaluate a function at a given point in its domain. A2.A.1.3:Solve one-variable rational equations and check for extraneous solutions.

Teaching Materials

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