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Difficulty: Very hardFunction Evaluation, Composition, and Properties

Let the functions ff and gg be defined for all real numbers by f(x)=∣2x−6∣−16f(x) = |2x - 6| - 16 and g(x)=(x−3)2−5g(x) = (x - 3)^2 - 5. What is the sum of all real values of xx that satisfy the equation f(g(x))=0f(g(x)) = 0?

  1. A
    0
  2. B
    6
  3. 9Answer
  4. D
    10
  5. E
    12

Answer

9
The correct answer is 9. First, substitute the inner function into the outer function to get f(g(x))=∣2g(x)−6∣−16=0f(g(x)) = |2g(x) - 6| - 16 = 0. This simplifies to the absolute value equation ∣2g(x)−6∣=16|2g(x) - 6| = 16. Splitting this into its two possible cases gives 2g(x)−6=16  ⟹  g(x)=112g(x) - 6 = 16 \implies g(x) = 11, and 2g(x)−6=−16  ⟹  g(x)=−52g(x) - 6 = -16 \implies g(x) = -5. Solving the first case, (x−3)2−5=11  ⟹  (x−3)2=16(x-3)^2 - 5 = 11 \implies (x-3)^2 = 16. Taking both square roots yields x−3=4  ⟹  x=7x - 3 = 4 \implies x = 7 and x−3=−4  ⟹  x=−1x - 3 = -4 \implies x = -1. Solving the second case, (x−3)2−5=−5  ⟹  (x−3)2=0  ⟹  x−3=0  ⟹  x=3(x-3)^2 - 5 = -5 \implies (x-3)^2 = 0 \implies x - 3 = 0 \implies x = 3. The sum of all unique real values of xx that satisfy the original equation is 7+(−1)+3=97 + (-1) + 3 = 9.

Step-by-Step Solution

1
Set up the composite function equation using the outer function definition.
∣2g(x)−6∣−16=0|2g(x) - 6| - 16 = 0
Substitute the expression of g(x)g(x) as the input variable into f(x)f(x).
2
Isolate the absolute value term and split the equation into two cases.
∣2g(x)−6∣=16  ⟹  2g(x)−6=16|2g(x) - 6| = 16 \implies 2g(x) - 6 = 16 or 2g(x)−6=−162g(x) - 6 = -16
An absolute value equation of the form ∣A∣=B|A| = B where B>0B > 0 has two solutions: A=BA = B and A=−BA = -B.
3
Solve for the target values of g(x)g(x) in each case.
g(x)=11g(x) = 11 and g(x)=−5g(x) = -5
Isolate the function g(x)g(x) by adding 6 and dividing by 2 on both sides of each equation.
4
Solve the first case g(x)=11g(x) = 11 for the variable xx.
(x−3)2−5=11  ⟹  (x−3)2=16  ⟹  x−3=±4(x-3)^2 - 5 = 11 \implies (x-3)^2 = 16 \implies x - 3 = \pm 4, yielding x=7x = 7 and x=−1x = -1
Substitute the algebraic rule for g(x)g(x), isolate the squared expression, and extract the square roots.
5
Solve the second case g(x)=−5g(x) = -5 for the variable xx.
(x−3)2−5=−5  ⟹  (x−3)2=0  ⟹  x−3=0(x-3)^2 - 5 = -5 \implies (x-3)^2 = 0 \implies x - 3 = 0, yielding x=3x = 3
Substitute the algebraic rule for g(x)g(x), isolate the squared expression, and solve for xx.
6
Sum all unique real solutions found.
7+(−1)+3=97 + (-1) + 3 = 9
Add the distinct solutions x=7x = 7, x=−1x = -1, and x=3x = 3 to find the total sum.

Key Concept

Evaluating and solving equations involving composite functions, absolute values, and quadratic expressions.
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Topics

Function Evaluation, Composition, and Properties

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