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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 two functions \(f(x)=\frac{3}{x-4}\) and \(g(x)=\frac{3}{x}+4\) are inverses of each other. After calculation, both expressions \(f(g(x))\) and \(g(f(x))\) resulted in \(x\).

Step by step solution

01

Find f(g(x))

Plug \(g(x)\) into \(f(x)\). So wherever there is \(x\) in the function \(f(x)\), replace it with the function \(g(x)\). So we replace \(x\) with \(\frac{3}{x}+4\) to get: \(f(g(x)) = \frac{3}{\left(\frac{3}{x}+4\right)-4}\)
02

Simplify f(g(x))

Now simplify the function \(f(g(x))\). If the denominator is a fraction, it can be multiplied by its reciprocal to simplify. Therefore the equation becomes: \(f(g(x)) = \frac{3}{\frac{3}{x}} = x\)
03

Find g(f(x))

Plug \(f(x)\) into \(g(x)\). So wherever there is \(x\) in the function \(g(x)\), replace it with the function \(f(x)\). So we replace \(x\) with \(\frac{3}{x-4}\) to get: \(g(f(x)) = \frac{3}{\left(\frac{3}{x-4}\right)}+4\)
04

Simplify g(f(x))

Now simplify the function \(g(f(x))\). If the denominator is a fraction, it can be multiplied by its reciprocal to simplify. Therefore the equation becomes: \(g(f(x)) = \frac{3}{\frac{3}{x-4}} + 4 = x+4-4 = x\)
05

Determine if f and g are inverses

Both \(f(g(x))\) and \(g(f(x))\) simplify to \(x\). That means that now we know that these two functions, \(f\) and \(g\), are indeed inverses of each other.

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