Function Composition: Mastering f(g(x)) and g(f(x))

PreAlgebra Grades High School 5:43 Video
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Lesson Description

Explore the concept of function composition, learn how to evaluate composite functions, and understand the importance of order in function composition. This lesson uses examples from Mario's Math Tutoring to build your precalculus skills.

Video Resource

Composition of Functions - How to

Mario's Math Tutoring

Duration: 5:43
Watch on YouTube

Key Concepts

  • Function Notation
  • Composition of Functions
  • Order of Operations in Composition
  • Domain Restrictions

Learning Objectives

  • Students will be able to evaluate composite functions for given values.
  • Students will be able to determine the algebraic expression for a composite function f(g(x)) or g(f(x)).
  • Students will be able to understand that f(g(x)) is generally not equal to g(f(x)).
  • Students will be able to identify domain restrictions in composed functions.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing function notation and how to evaluate a function for a given input. For example, if f(x) = 2x - 3, demonstrate how to find f(3), f(a), and f(7). Connect this to the idea of substitution. Link this to the idea of one step calculations and then introduce composition as a way to accomplish two functions at once.
  • What is Composition? (10 mins)
    Explain the concept of function composition using the notation f(g(x)). Illustrate how g(x) acts as the input for the function f(x). Provide a concrete example, such as f(x) = 2x - 3 and g(x) = x^2 + 1. Evaluate f(g(2)) step-by-step: first find g(2), then substitute that result into f(x). Follow along with the example in the video from 0:51 - 4:17.
  • Finding the Composite Function Algebraically (15 mins)
    Demonstrate how to find the expression for f(g(x)) without evaluating at a specific point. Substitute the expression for g(x) into f(x). Simplify the resulting expression. Emphasize that this results in a new function that performs both operations at once. Reiterate the previous example to show how this method can be used to solve for f(g(2)). Follow along with the example in the video from 0:51 - 4:17.
  • Order Matters! (10 mins)
    Explain that f(g(x)) is generally not the same as g(f(x)). Demonstrate how to find g(f(x)) using the same example functions as before. Compare the resulting expression to the expression for f(g(x)). Stress the importance of paying attention to the order of composition. Follow along with the example in the video from 0:51 - 4:17.
  • Alternative Notation and Advanced Example (15 mins)
    Introduce the alternative notation (f ∘ g)(x) for function composition and explain that it means the same as f(g(x)). Work through a more complex example, such as composing h(x) = 3/(x-4) and f(x) = 2x-3 to find h(f(x)). Simplify the resulting expression. Address potential domain restrictions, such as values of x that would make the denominator zero. Follow along with the example in the video from 4:17 - end.
  • Wrap-up and Review (5 mins)
    Summarize the key concepts of function composition. Remind students to pay attention to the order of composition and potential domain restrictions. Encourage students to practice with additional examples.

Interactive Exercises

  • Composition Practice
    Provide students with several pairs of functions, f(x) and g(x), and ask them to find both f(g(x)) and g(f(x)). Include examples with different types of functions (linear, quadratic, rational) to provide varied practice. Then, have students find a specific value f(g(3)) or g(f(-1)) for each pair of functions.
  • Domain Restriction Challenge
    Provide students with function composition problems involving rational functions where they have to identify and explain any domain restrictions of the composite function.

Discussion Questions

  • Why is the order of functions important in composition?
  • What are some real-world examples of function composition?
  • How can domain restrictions affect the composition of functions?

Skills Developed

  • Algebraic manipulation
  • Analytical thinking
  • Problem-solving

Multiple Choice Questions

Question 1:

Given f(x) = x + 2 and g(x) = x^2, what is f(g(x))?

Correct Answer: x^2 + 2

Question 2:

If h(x) = 2x - 1 and k(x) = √x, what is h(k(4))?

Correct Answer: 3

Question 3:

Given p(x) = 1/x and q(x) = x - 3, what is the domain restriction for p(q(x))?

Correct Answer: x ≠ 3

Question 4:

If f(x) = x^2 and g(x) = x + 1, which of the following is true?

Correct Answer: f(g(x)) ≠ g(f(x))

Question 5:

Let f(x) = 3x + 2 and g(x) = |x|. What is f(g(-2))?

Correct Answer: 8

Question 6:

If r(x) = x/(x-1) and s(x) = x + 2, what is r(s(x))?

Correct Answer: (x+2)/(x+1)

Question 7:

Which notation is equivalent to f(g(x))?

Correct Answer: (f ∘ g)(x)

Question 8:

If a(x) = x^3 and b(x) = x - 1, what is b(a(2))?

Correct Answer: 7

Question 9:

If f(x) = x + 5 and g(x) = √x, for what values of x is g(f(x)) defined?

Correct Answer: x ≥ -5

Question 10:

If m(x) = 4 - x and n(x) = x^2 + 1, what is m(n(x))?

Correct Answer: 3 - x^2

Fill in the Blank Questions

Question 1:

Given f(x) = 2x + 1 and g(x) = x - 4, then f(g(x)) = ______.

Correct Answer: 2x-7

Question 2:

If h(x) = x^2 + 3 and k(x) = √x, then h(k(x)) = ______.

Correct Answer: x+3

Question 3:

The alternative notation for f(g(x)) is (f ∘ g)(__).

Correct Answer: x

Question 4:

If p(x) = 5x and q(x) = x/2, then q(p(4)) = ______.

Correct Answer: 10

Question 5:

Given r(x) = 1/(x + 1), the domain restriction for r(r(x)) is x cannot equal ______.

Correct Answer: -2

Question 6:

If s(x) = x - 2 and t(x) = x^2, then t(s(x)) = ______.

Correct Answer: (x-2)^2

Question 7:

For functions f(x) and g(x), the order of composition ______ matter.

Correct Answer: does

Question 8:

If a(x) = x + 7 and b(x) = 3x, then a(b(x)) = ______.

Correct Answer: 3x+7

Question 9:

When composing functions involving square roots, it's important to consider the ______ of the resulting function.

Correct Answer: domain

Question 10:

If m(x) = x/(x-5), the domain restriction of m(x) is x cannot equal ____.

Correct Answer: 5

Educational Standards

PC.F.2.1:Model relationships through composition, and attend to the restrictions of the domain. PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation. PC.F.3.3:Algebraically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic.

Teaching Materials

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