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Difficulty: MediumFunction Evaluation, Composition, and Properties

Two functions, ff and gg, are defined as f(x)=(x−4)2f(x) = (x - 4)^2 and g(x)=∣2x−3∣g(x) = |2x - 3|. What is the value of the composite function f(g(x))f(g(x)) evaluated at x=−1x = -1?

  1. 1Answer
  2. B
    9
  3. C
    47
  4. D
    81
  5. E
    125

Answer

1
To find f(g(−1))f(g(-1)), we first evaluate the inner function gg at x=−1x = -1. Substituting −1-1 into g(x)=∣2x−3∣g(x) = |2x - 3| gives g(−1)=∣2(−1)−3∣=∣−5∣=5g(-1) = |2(-1) - 3| = |-5| = 5. Next, we use this result as the input for the outer function ff, evaluating f(5)f(5). Substituting 55 into f(x)=(x−4)2f(x) = (x - 4)^2 gives f(5)=(5−4)2=12=1f(5) = (5 - 4)^2 = 1^2 = 1. This corresponds to the option with value 1.

Step-by-Step Solution

1
Evaluate the inner function g(−1)g(-1) first.
g(−1)=∣2(−1)−3∣=∣−2−3∣=∣−5∣=5g(-1) = |2(-1) - 3| = |-2 - 3| = |-5| = 5
The input to the outer function of a composition is the output of the inner function.
2
Evaluate the outer function f(x)f(x) at the result from the previous step.
f(5)=(5−4)2=12=1f(5) = (5 - 4)^2 = 1^2 = 1
We substitute the value of g(−1)g(-1), which is 55, into the function f(x)f(x) to find the final value of the composition.

Key Concept

Evaluating a composite function involves applying the inner function first, then using that output as the input for the outer function.
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Topics

Function Evaluation, Composition, and Properties

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