Math Problem Statement

Quiz:Chapter 4 Skills Check Quiz (Integrated Review) Question 1 of 4 This quiz: 4 point(s) possible This question: 1 point(s) possible

Skip to Main content Question 1 Question 2 Question 3 Question 4 Question content area top Part 1 Let ​f(x)equalsStartAbsoluteValue x EndAbsoluteValue​, ​g(x)equalsxminus2​, and ​h(x)equalsx squared. Write ​N(x)equalsxminus4 as a composition of functions. Question content area bottom Part 1 Choose the following composition that correctly defines ​N(x)equalsxminus4. A. gcircleg B. fcirclegcircleh C. gcircleh D. hcircleg Choose the following composition that correctly defines N(x)x4.

Solution

To write N(x)=x4N(x) = x - 4 as a composition of functions, let's analyze the options.

Given Functions:

  1. f(x)=xf(x) = |x|
  2. g(x)=x2g(x) = x - 2
  3. h(x)=x2h(x) = x^2

We need to form N(x)=x4N(x) = x - 4 using these functions.

Composition Analysis:

Option A: ggg \circ g

  • gg(x)=g(g(x))g \circ g(x) = g(g(x))
  • g(x)=x2g(x) = x - 2, so g(g(x))=(x2)2=x4g(g(x)) = (x - 2) - 2 = x - 4.
  • This matches N(x)=x4N(x) = x - 4.

Option B: fghf \circ g \circ h

  • This involves applying h(x)=x2h(x) = x^2, g(x)=x2g(x) = x - 2, and f(x)=xf(x) = |x|, which does not result in N(x)=x4N(x) = x - 4.
  • This does not match N(x)N(x).

Option C: ghg \circ h

  • g(h(x))=g(x2)=x22g(h(x)) = g(x^2) = x^2 - 2, which is not N(x)=x4N(x) = x - 4.
  • This does not match N(x)N(x).

Option D: hgh \circ g

  • h(g(x))=h(x2)=(x2)2h(g(x)) = h(x - 2) = (x - 2)^2, which is not N(x)=x4N(x) = x - 4.
  • This does not match N(x)N(x).

Correct Answer:

A. ggg \circ g


Would you like me to explain further or assist with additional concepts?

Relative Questions:

  1. How do you compose functions step by step?
  2. What are the common properties of function compositions?
  3. Can compositions include transformations like reflection or scaling?
  4. What happens when absolute value functions are composed with other types of functions?
  5. How do quadratic functions behave when part of a composition?

Tip: Always check the intermediate steps in a composition to ensure the final function matches the target expression.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Linear Functions

Formulas

Composition of Functions: (f ∘ g)(x) = f(g(x))

Theorems

Properties of Function Composition

Suitable Grade Level

Grades 9-11