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Difficulty: HardFunction Evaluation, Composition, and Properties

For the functions f(x)=x2−3x−1f(x) = x^2 - 3x - 1 and g(x)=∣x−1∣g(x) = |x - 1|, what is the product of all real values of xx for which f(g(x))=3f(g(x)) = 3?

  1. A
    5
  2. B
    0
  3. -15Answer
  4. D
    -5
  5. E
    -6

Answer

The product of all real values of xx is −15-15.
The correct answer is −15-15. By substituting g(x)=∣x−1∣g(x) = |x - 1| into f(x)=x2−3x−1f(x) = x^2 - 3x - 1, we get ∣x−1∣2−3∣x−1∣−1=3|x - 1|^2 - 3|x - 1| - 1 = 3. Letting u=∣x−1∣u = |x - 1| yields the quadratic u2−3u−4=0u^2 - 3u - 4 = 0, which factors into (u−4)(u+1)=0(u - 4)(u + 1) = 0. Since u≥0u \geq 0, we have u=4u = 4. Solving ∣x−1∣=4|x - 1| = 4 yields x=5x = 5 and x=−3x = -3. The product of these solutions is 5×(−3)=−155 \times (-3) = -15.

Step-by-Step Solution

1
Substitute the function g(x)=∣x−1∣g(x) = |x - 1| into f(x)=x2−3x−1f(x) = x^2 - 3x - 1 to form the composite function f(g(x))f(g(x)).
f(g(x))=∣x−1∣2−3∣x−1∣−1f(g(x)) = |x - 1|^2 - 3|x - 1| - 1
To evaluate the composite function before setting it equal to 33.
2
Set the composite function equal to 33 and substitute u=∣x−1∣u = |x - 1| (where u≥0u \geq 0) to rewrite the equation as a quadratic in terms of uu.
u2−3u−1=3⇒u2−3u−4=0u^2 - 3u - 1 = 3 \Rightarrow u^2 - 3u - 4 = 0
Using substitution simplifies the absolute value equation into a standard quadratic equation.
3
Factor the quadratic equation (u−4)(u+1)=0(u - 4)(u + 1) = 0 to solve for uu, and discard the extraneous solution.
u=4u = 4 or u=−1u = -1. Since u=∣x−1∣≥0u = |x - 1| \geq 0, we discard u=−1u = -1, leaving u=4u = 4.
The absolute value of a real expression must be non-negative, so uu cannot be −1-1.
4
Solve the absolute value equation ∣x−1∣=4|x - 1| = 4 for all possible real values of xx.
x−1=4⇒x=5x - 1 = 4 \Rightarrow x = 5, and x−1=−4⇒x=−3x - 1 = -4 \Rightarrow x = -3.
An absolute value equation ∣A∣=B|A| = B (where B>0B > 0) yields two scenarios: A=BA = B and A=−BA = -B.
5
Calculate the product of the two real solutions found.
5×(−3)=−155 \times (-3) = -15
To find the final product of all real values of xx that satisfy the equation.

Key Concept

Function composition and solving absolute value equations using quadratic substitution
Estimated Time:2m 0s
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Topics

Function Evaluation, Composition, and Properties

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