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Difficulty: MediumFunction Evaluation, Composition, and Properties

Let the functions ff and gg be defined by f(x)=3x+4f(x) = \sqrt{3x + 4} and g(x)=x2−2g(x) = x^2 - 2. What is the positive value of xx for which the composite function value f(g(x))=5f(g(x)) = 5?

Answer: 3

Answer

3
The correct answer is 3. We set f(g(x))=5f(g(x)) = 5, which translates to 3g(x)+4=5\sqrt{3g(x) + 4} = 5. Squaring both sides yields 3g(x)+4=253g(x) + 4 = 25. Solving for g(x)g(x) gives 3g(x)=213g(x) = 21, which simplifies to g(x)=7g(x) = 7. Substituting g(x)=x2−2g(x) = x^2 - 2 into this equation gives x2−2=7x^2 - 2 = 7, which simplifies to x2=9x^2 = 9. Taking the square root of both sides gives x=3x = 3 or x=−3x = -3. Since the question specifies the positive value of xx, the correct answer is 3.

Step-by-Step Solution

1
Substitute the inner function g(x)g(x) into the outer function f(x)f(x) to express f(g(x))=5f(g(x)) = 5
3g(x)+4=5\sqrt{3g(x) + 4} = 5
Setting up the initial composite equation to isolate the inner function g(x)g(x)
2
Square both sides of the equation and isolate g(x)g(x)
3g(x)+4=25  ⟹  3g(x)=21  ⟹  g(x)=73g(x) + 4 = 25 \implies 3g(x) = 21 \implies g(x) = 7
Eliminating the radical allows us to determine the required output value of g(x)g(x)
3
Substitute the formula for g(x)g(x) into the equation and solve for xx
x2−2=7  ⟹  x2=9  ⟹  x=3x^2 - 2 = 7 \implies x^2 = 9 \implies x = 3 or x=−3x = -3
Finding the input values of xx that satisfy the condition for g(x)g(x)
4
Select the positive value from the solution set
x=3x = 3
The question specifically requests the positive value of xx

Key Concept

Evaluating and solving composite function equations by working from the outside function inward

Alternative Method

Instead of solving from the outside in, we can first construct the symbolic expression for f(g(x))f(g(x)) by substituting g(x)g(x) directly: f(g(x))=3(x2−2)+4=3x2−6+4=3x2−2f(g(x)) = \sqrt{3(x^2 - 2) + 4} = \sqrt{3x^2 - 6 + 4} = \sqrt{3x^2 - 2}. Setting this equal to 5 yields 3x2−2=5\sqrt{3x^2 - 2} = 5. Squaring both sides gives 3x2−2=25  ⟹  3x2=27  ⟹  x2=9  ⟹  x=±33x^2 - 2 = 25 \implies 3x^2 = 27 \implies x^2 = 9 \implies x = \pm 3. The positive solution is 3.
Estimated Time:1m 30s
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Topics

Function Evaluation, Composition, and Properties

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