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Difficulty: MediumFunction Evaluation, Composition, and Properties

The functions ff and gg are defined for all permissible real numbers by f(x)=x+3x−1f(x) = \frac{x + 3}{x - 1} and g(x)=2x−5g(x) = 2x - 5. If (f∘g)(x)=3(f \circ g)(x) = 3, what is the value of xx?

Answer: 4

Answer

The value of xx that satisfies the equation (f∘g)(x)=3(f \circ g)(x) = 3 is 44.
To solve for xx in (f∘g)(x)=3(f \circ g)(x) = 3, we find the composite function f(g(x))f(g(x)) by substituting g(x)=2x−5g(x) = 2x - 5 into f(x)f(x). This yields f(g(x))=(2x−5)+3(2x−5)−1=2x−22x−6f(g(x)) = \frac{(2x-5)+3}{(2x-5)-1} = \frac{2x-2}{2x-6}. Setting this equal to 33 gives 2x−22x−6=3\frac{2x-2}{2x-6} = 3. Multiplying by 2x−62x-6 yields 2x−2=6x−182x-2 = 6x-18. Rearranging terms to isolate xx gives 4x=164x = 16, which results in x=4x = 4.

Step-by-Step Solution

1
Substitute the expression for g(x)g(x) into f(x)f(x) to obtain the composite function (f∘g)(x)(f \circ g)(x).
(f∘g)(x)=2x−22x−6(f \circ g)(x) = \frac{2x - 2}{2x - 6}
By definition of function composition, (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x)). Substituting g(x)=2x−5g(x) = 2x - 5 into f(x)=x+3x−1f(x) = \frac{x+3}{x-1} yields f(2x−5)=(2x−5)+3(2x−5)−1f(2x-5) = \frac{(2x-5)+3}{(2x-5)-1}, which simplifies to 2x−22x−6\frac{2x-2}{2x-6}.
2
Set the composite function expression equal to 33 and solve for xx.
x=4x = 4
We set 2x−22x−6=3\frac{2x-2}{2x-6} = 3. Multiplying both sides by the denominator 2x−62x-6 gives 2x−2=3(2x−6)2x-2 = 3(2x-6). Expanding the right side gives 2x−2=6x−182x-2 = 6x-18. Subtracting 2x2x from both sides and adding 1818 to both sides results in 16=4x16 = 4x. Dividing by 44 gives x=4x = 4.

Key Concept

Function Composition and Evaluation
Estimated Time:1m 30s
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Topics

Function Evaluation, Composition, and Properties

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