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Difficulty: HardFunction Evaluation, Composition, and Properties

For the functions f(x)=(x−3)2−4f(x) = (x - 3)^2 - 4 and g(x)=∣x+1∣g(x) = |x + 1|, what is the sum of all real values of xx for which f(g(x))=5f(g(x)) = 5?

  1. −3-3Answer
  2. B
    −2-2
  3. C
    44
  4. D
    55
  5. E
    66

Answer

The sum of all real values of xx for which f(g(x))=5f(g(x)) = 5 is −3-3.
To find the sum of all real values of xx for which f(g(x))=5f(g(x)) = 5, we set up the composite function: (g(x)−3)2−4=5(g(x) - 3)^2 - 4 = 5. Adding 44 to both sides gives (g(x)−3)2=9(g(x) - 3)^2 = 9. Taking the square root yields two possible cases: g(x)−3=3g(x) - 3 = 3 (which means g(x)=6g(x) = 6) or g(x)−3=−3g(x) - 3 = -3 (which means g(x)=0g(x) = 0). Next, we substitute g(x)=∣x+1∣g(x) = |x + 1|. For the first case, ∣x+1∣=6|x + 1| = 6 gives x+1=6  ⟹  x=5x + 1 = 6 \implies x = 5 or x+1=−6  ⟹  x=−7x + 1 = -6 \implies x = -7. For the second case, ∣x+1∣=0|x + 1| = 0 gives x+1=0  ⟹  x=−1x + 1 = 0 \implies x = -1. Summing these three real solutions gives 5+(−7)+(−1)=−35 + (-7) + (-1) = -3.

Step-by-Step Solution

1
Substitute g(x)g(x) into f(x)f(x) and set the expression equal to 55.
f(g(x))=(g(x)−3)2−4=5f(g(x)) = (g(x) - 3)^2 - 4 = 5
This sets up the composition equation to be solved.
2
Isolate the squared term and solve for g(x)g(x) by taking the square root of both sides.
(g(x)−3)2=9(g(x) - 3)^2 = 9, which gives two cases: g(x)−3=3  ⟹  g(x)=6g(x) - 3 = 3 \implies g(x) = 6 or g(x)−3=−3  ⟹  g(x)=0g(x) - 3 = -3 \implies g(x) = 0
Solving a quadratic equation of the form u2=ku^2 = k yields two possibilities: u=±ku = \pm\sqrt{k}.
3
Substitute the expression for g(x)=∣x+1∣g(x) = |x + 1| into both cases and solve for xx.
For g(x)=6g(x) = 6, ∣x+1∣=6  ⟹  x+1=6|x + 1| = 6 \implies x + 1 = 6 or x+1=−6x + 1 = -6, yielding x=5x = 5 or x=−7x = -7. For g(x)=0g(x) = 0, ∣x+1∣=0  ⟹  x+1=0|x + 1| = 0 \implies x + 1 = 0, yielding x=−1x = -1.
Absolute value equations of the form ∣v∣=c|v| = c yield v=±cv = \pm c if c>0c > 0, and v=0v = 0 if c=0c = 0.
4
Calculate the sum of all the real solutions.
5+(−7)+(−1)=−35 + (-7) + (-1) = -3
The question asks for the sum of all real values of xx that satisfy the equation.

Key Concept

Function composition involves substituting one function into another, and solving the resulting equation requires accounting for multiple cases when dealing with quadratic and absolute value expressions.
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Topics

Function Evaluation, Composition, and Properties

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