Function Fusion: Mastering Composition of Functions

Algebra 2 Grades High School 14:48 Video
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Lesson Description

Learn how to combine functions through composition, understanding notation like f(g(x)), and determining domains of composite functions.

Video Resource

Composition of Functions

Mario's Math Tutoring

Duration: 14:48
Watch on YouTube

Key Concepts

  • Composition of functions
  • f(g(x)) notation
  • Domain of composite functions
  • Evaluating composite functions

Learning Objectives

  • Students will be able to evaluate composite functions given specific numerical inputs.
  • Students will be able to determine the composite function f(g(x)) and g(f(x)) given two functions f(x) and g(x).
  • Students will be able to identify the domain of a composite function.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of functions and function notation. Briefly explain that this lesson will introduce a way to combine functions called composition. Show the video to the class.
  • Understanding Composition Notation (10 mins)
    After the video, discuss the f(g(x)) notation and its meaning. Emphasize that it means 'f of g of x' and that you're plugging the entire function g(x) into the function f(x). Reinforce the concept of working from the inside out.
  • Numerical Evaluation Examples (15 mins)
    Work through the numerical examples from the video (like finding f(g(2))). Stress the step-by-step process: first, evaluate the inner function (g(2)), then use that result as the input for the outer function (f(x)). Provide additional similar examples for students to practice.
  • Finding the Composite Function (15 mins)
    Explain how to find the composite function f(g(x)) as a new function. Reiterate that you are substituting the entire g(x) expression into the x of the f(x) function. Go through the algebraic steps of substituting and simplifying. Also, find the composite function g(f(x)) as a new function. Walk through the algebraic steps of substituting and simplifying.
  • Determining the Domain of Composite Functions (15 mins)
    Emphasize the importance of considering the domain *before* simplifying the composite function. Review restrictions on domains, such as no division by zero and no square roots of negative numbers. Work through the domain examples from the video, and provide additional practice problems focusing on finding the domain of f(g(x)).
  • Practice Problems (10 mins)
    Have students work independently or in pairs on practice problems involving both numerical evaluation and finding the composite function and its domain. Circulate to provide assistance.

Interactive Exercises

  • Function Machine
    Create a 'function machine' activity where students are given two functions and a set of inputs. They must first evaluate the inner function with the input and then use the result as the input for the outer function, showing the step-by-step process.
  • Domain Challenge
    Present students with several composite functions and challenge them to determine the domain, explaining their reasoning. This can be done as a small group activity with students defending their answers.

Discussion Questions

  • Why is it important to consider the domain before simplifying a composite function?
  • How does the order of composition affect the resulting function (i.e., is f(g(x)) always the same as g(f(x)))?
  • Can you provide a real-world example of function composition?

Skills Developed

  • Function evaluation
  • Algebraic manipulation
  • Domain identification
  • Problem-solving

Multiple Choice Questions

Question 1:

What does f(g(x)) mean?

Correct Answer: f of g of x, meaning g(x) is the input for f(x)

Question 2:

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

Correct Answer: 11

Question 3:

If h(x) = √x and k(x) = x - 1, what is the domain of h(k(x))?

Correct Answer: x ≥ 1

Question 4:

Given f(x) = 2x - 1 and g(x) = x / 3, find f(g(x)).

Correct Answer: 2x/3 - 1

Question 5:

When finding the domain of a composite function, why is it important to check the domain restrictions before simplifying the expression?

Correct Answer: Simplifying can mask domain restrictions that were present in the original functions.

Question 6:

If f(x) = x² + 1, what is f(f(x))?

Correct Answer: x⁴ + 2x² + 2

Question 7:

Which of the following represents the correct order of operations when evaluating f(g(a))?

Correct Answer: Evaluate g(a) first, then substitute the result into f(x).

Question 8:

If m(x) = 1/(x-2), what value(s) must be excluded from the domain of m(m(x))?

Correct Answer: x ≠ 2 and x ≠ 3

Question 9:

Given the graphs of f(x) and g(x), how do you find f(g(2))?

Correct Answer: Find the y-value of g(x) when x = 2, then find the y-value of f(x) when x equals that y-value.

Question 10:

If p(x) = |x| and q(x) = x - 3, what is p(q(x))?

Correct Answer: |x - 3|

Fill in the Blank Questions

Question 1:

The notation f(g(x)) means you are composing function g with function ____.

Correct Answer: f

Question 2:

When evaluating f(g(5)), you first evaluate g(____).

Correct Answer: 5

Question 3:

Before simplifying a composite function, you must consider its ____.

Correct Answer: domain

Question 4:

If f(x) = x + 1 and g(x) = x², then g(f(x)) = (x + ____)².

Correct Answer: 1

Question 5:

A value is excluded from the domain of a composite function if it causes division by ____ or the square root of a negative number.

Correct Answer: zero

Question 6:

The domain of a composite function is the set of all x-values that are valid inputs for ____.

Correct Answer: g(x)

Question 7:

If f(x) = 3x and g(x) = x - 2, then f(g(x)) = 3x - ____.

Correct Answer: 6

Question 8:

When composing functions represented graphically, you use the ____ of the inner function as the input for the outer function.

Correct Answer: y-value

Question 9:

The function that is 'inside' the other function in the notation f(g(x)) is g(____).

Correct Answer: x

Question 10:

If f(x) = √(x) and g(x) = x+3, the domain of f(g(x)) is x ≥ ____.

Correct Answer: -3

Educational Standards

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.2.5:Rewrite algebraic expressions involving radicals and rational exponents using the properties of exponents.

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