Examkin

Question

Difficulty: MediumFunction Evaluation, Composition, and Properties

For the functions f(x)=∣x−2∣−3f(x) = |x - 2| - 3 and g(x)=x2−2xg(x) = x^2 - 2x, what is the sum of all real values of xx for which g(f(x))=0g(f(x)) = 0?

  1. A
    2
  2. B
    4
  3. 8Answer
  4. D
    12
  5. E
    -8

Answer

The sum of all real values of xx is 8.
To solve g(f(x))=0g(f(x)) = 0, we substitute f(x)f(x) as the variable in g(x)g(x), obtaining (f(x))2−2f(x)=0(f(x))^2 - 2f(x) = 0. Factoring this equation yields f(x)(f(x)−2)=0f(x)(f(x) - 2) = 0, giving two cases: f(x)=0f(x) = 0 or f(x)=2f(x) = 2. For the first case, ∣x−2∣−3=0|x - 2| - 3 = 0 simplifies to ∣x−2∣=3|x - 2| = 3, which has solutions x=5x = 5 and x=−1x = -1. For the second case, ∣x−2∣−3=2|x - 2| - 3 = 2 simplifies to ∣x−2∣=5|x - 2| = 5, which has solutions x=7x = 7 and x=−3x = -3. Summing these four solutions gives 5+(−1)+7+(−3)=85 + (-1) + 7 + (-3) = 8.

Step-by-Step Solution

1
Substitute f(x)f(x) into g(x)g(x) to set up the composite equation g(f(x))=0g(f(x)) = 0.
g(f(x))=(f(x))2−2f(x)=0g(f(x)) = (f(x))^2 - 2f(x) = 0
To evaluate a function composition, the output of the inner function f(x)f(x) becomes the input of the outer function g(x)g(x).
2
Factor the quadratic equation in terms of f(x)f(x).
f(x)(f(x)−2)=0f(x)(f(x) - 2) = 0, which means f(x)=0f(x) = 0 or f(x)=2f(x) = 2.
Factoring allows us to split the quadratic equation into two separate linear cases.
3
Substitute f(x)=∣x−2∣−3f(x) = |x - 2| - 3 into the first case, f(x)=0f(x) = 0, and solve for xx.
∣x−2∣−3=0⇒∣x−2∣=3⇒x−2=3|x - 2| - 3 = 0 \Rightarrow |x - 2| = 3 \Rightarrow x - 2 = 3 or x−2=−3x - 2 = -3, yielding x=5x = 5 and x=−1x = -1.
An absolute value equation ∣u∣=c|u| = c has two cases: u=cu = c and u=−cu = -c when c≥0c \geq 0.
4
Substitute f(x)=∣x−2∣−3f(x) = |x - 2| - 3 into the second case, f(x)=2f(x) = 2, and solve for xx.
∣x−2∣−3=2⇒∣x−2∣=5⇒x−2=5|x - 2| - 3 = 2 \Rightarrow |x - 2| = 5 \Rightarrow x - 2 = 5 or x−2=−5x - 2 = -5, yielding x=7x = 7 and x=−3x = -3.
We must solve the second case using both positive and negative scenarios for the absolute value.
5
Calculate the sum of all four real solutions.
5+(−1)+7+(−3)=85 + (-1) + 7 + (-3) = 8
The question asks for the sum of all real values of xx that satisfy the composite equation.

Key Concept

Function Composition and Absolute Value Equations

Alternative Method

Instead of expanding the algebraic expressions, substitute a temporary variable u=f(x)u = f(x) into g(u)=0g(u) = 0 to get u2−2u=0u^2 - 2u = 0, which easily factors to u=0u = 0 or u=2u = 2. Then, substitute f(x)f(x) back to solve ∣x−2∣−3=0|x - 2| - 3 = 0 and ∣x−2∣−3=2|x - 2| - 3 = 2.
Estimated Time:1m 30s
Rate this question

Topics

Function Evaluation, Composition, and Properties

More questions from this topic