Function Fusion: Mastering Composition of Functions

Algebra 2 Grades High School 7:07 Video
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Lesson Description

Explore the world of composite functions, understand different notations, and learn how to evaluate and simplify them. This lesson covers both numerical evaluation and algebraic composition.

Video Resource

How to do Composition of Functions

Mario's Math Tutoring

Duration: 7:07
Watch on YouTube

Key Concepts

  • Composition of functions
  • Function notation (f(g(x)) and (f ∘ g)(x))
  • Order of operations in function composition
  • Double substitution method
  • Evaluating composite functions numerically
  • Evaluating composite functions algebraically
  • Simplifying composite functions

Learning Objectives

  • Students will be able to evaluate composite functions at a given point.
  • Students will be able to simplify composite functions algebraically.
  • Students will be able to understand and apply different notations for composite functions.
  • Students will be able to explain the difference between f(g(x)) and g(f(x)).

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the basic concept of functions and function notation. Introduce the idea of 'functions within functions' and explain that this lesson will delve into composition of functions. Briefly discuss the two notations used: f(g(x)) and (f ∘ g)(x).
  • Numerical Evaluation of Composite Functions (15 mins)
    Using the video's first two examples as a guide, demonstrate how to evaluate composite functions numerically. Emphasize the 'double substitution' method. Work through f(g(2)) and g(f(2)), highlighting the importance of order. Provide additional similar examples for students to practice.
  • Algebraic Composition of Functions (20 mins)
    Transition to algebraic composition, where the input is an expression rather than a number. Follow the video's examples of finding f(g(x)) and g(f(x)) when f(x) and g(x) are given as algebraic expressions. Stress the importance of correct substitution and simplification. Show how evaluating the composite function at a number yields the same result as the double substitution method. Provide practice problems with varying complexity.
  • Practice and Application (15 mins)
    Provide a worksheet or online exercise with a variety of composition problems, including both numerical and algebraic examples. Encourage students to work independently or in pairs. Circulate to provide assistance and answer questions.
  • Wrap-up and Discussion (5 mins)
    Summarize the key concepts of function composition. Address any remaining questions or misconceptions. Preview upcoming topics related to functions.

Interactive Exercises

  • Drag and Drop Composition
    Provide students with functions f(x) and g(x) and ask them to drag and drop the correct expressions into the f(g(x)) and g(f(x)) templates. This can be done using an interactive whiteboard or online tool.
  • Online Function Composer
    Use an online tool (if available) where students can input functions and see the resulting composite function graphically and algebraically.

Discussion Questions

  • Why is the order important when composing functions (i.e., why is f(g(x)) generally not the same as g(f(x)))?
  • Can you think of real-world scenarios where composition of functions might be useful?
  • How does the domain and range of the individual functions affect the domain and range of the composite function?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Critical thinking
  • Attention to detail
  • Understanding of function notation

Multiple Choice Questions

Question 1:

What does (f ∘ g)(x) mean?

Correct Answer: f(g(x))

Question 2:

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

Correct Answer: 8

Question 3:

Given f(x) = x² and g(x) = x - 1, which expression represents f(g(x))?

Correct Answer: (x - 1)²

Question 4:

If f(x) = 3x and g(x) = x + 5, what is g(f(x))?

Correct Answer: 3x + 5

Question 5:

When evaluating f(g(x)), which function do you apply first?

Correct Answer: g(x)

Question 6:

What is the first step in finding f(g(2)) if f(x) = x^2 and g(x) = x + 1?

Correct Answer: Evaluate g(2) by substituting 2 into g(x).

Question 7:

Which of the following is true about f(g(x)) and g(f(x))?

Correct Answer: They are generally not equal.

Question 8:

If f(x) = x - 4 and g(x) = x^2, find g(f(x)).

Correct Answer: x^2 - 8x + 16

Question 9:

The process of plugging one function into another is called:

Correct Answer: Composition

Question 10:

If f(x) = 2x + 1 and g(x) = x/2, what is f(g(4))?

Correct Answer: 5

Fill in the Blank Questions

Question 1:

The notation (f ∘ g)(x) is read as 'f _____ g of x'.

Correct Answer: composed with

Question 2:

When evaluating a composite function, you start with the __________ function.

Correct Answer: innermost

Question 3:

If f(x) = x + 1 and g(x) = x², then f(g(x)) = __________.

Correct Answer: x² + 1

Question 4:

In the expression f(g(x)), g(x) is the __________ for f(x).

Correct Answer: input

Question 5:

The process of creating a composite function involves __________ one function into another.

Correct Answer: substituting

Question 6:

To find g(f(3)), first evaluate __________, then substitute that result into g(x).

Correct Answer: f(3)

Question 7:

If f(x) = 5x - 2 and g(x) = x + 3, then g(f(x)) = __________.

Correct Answer: 5x + 1

Question 8:

The order of functions matters in composition because f(g(x)) is generally __________ g(f(x)).

Correct Answer: not equal to

Question 9:

When simplifying composite functions, combine __________ __________ after substitution.

Correct Answer: like terms

Question 10:

If f(x) = x/3 and g(x) = x - 5, then f(g(x)) = __________.

Correct Answer: (x - 5)/3

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.2:Generate and evaluate equivalent algebraic expressions and equations using various strategies.

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