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Chapter 2: Problem 7

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=\frac{3}{x-4} \text { and } g(x)=\frac{3}{x}+4$$

Short Answer

Expert verified
The functions \(f\) and \(g\) are not inverses of each other. The composition of the two functions \(f(g(x))\) results in \(x\), but the composition \(g(f(x))\) does not result in \(x\), hence they are not inverses.

Step by step solution

01

Find the Composition \(f(g(x))\)

To find \(f(g(x))\), replace \(x\) in function \(f(x)\) with the function \(g(x)\). Thus, \(f(g(x)) = \frac{3}{{g(x)-4}} = \frac{3}{{\left(\frac{3}{x}+4\right)-4}}.\) Simplify this expression to find the result.
02

Simplify \(f(g(x))\)

Simplify \(f(g(x))\) by subtracting 4 in the denominator, and performing the multiplication operation. This is equals to \( \frac{3}{{\frac{3}{x}}} = x\).
03

Find the Composition \(g(f(x))\)

To find \(g(f(x))\), replace \(x\) in function \(g(x)\) with the function \(f(x)\). Thus, \(g(f(x)) = \frac{3}{{f(x)}}+4 = \frac{3}{{\frac{3}{{x-4}}}}+4.\) Simplify the expression to find the result.
04

Simplify \(g(f(x))\)

Simplify \(g(f(x))\) by multiplying the main fraction and adding 4. This equals to \(3(x-4)+4 = 3x-12+4 = 3x-8.\)
05

Determine if the Functions are Inverses

Check if \(f(g(x))=x\) and \(g(f(x))=y\). Since \(f(g(x)) = x\) but \(g(f(x)) \neq x\), it concludes that \(f\) and \(g\) are not inverses of each other.

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