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Chapter 2: Problem 5

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=5 x-9 \text { and } g(x)=\frac{x+5}{9}$$

Short Answer

Expert verified
The functions \(f\) and \(g\) are not inverses of each other as neither \(f(g(x))\) nor \(g(f(x))\) equals to \(x\).

Step by step solution

01

Find \(f(g(x))\)

Substitute \(g(x)\) into \(f(x)\) to get \(f(g(x))\). That is, substitute \(\frac{x+5}{9}\) into \(5x - 9\). The resulting equation is \(5(\frac{x+5}{9}) - 9\), which simplifies to \(\frac{x+5}{9}-9\).
02

Find \(g(f(x))\)

Substitute \(f(x)\) into \(g(x)\) to get \(g(f(x))\). Substitute \(5x-9\) into \(\frac{x+5}{9}\). The resulting equation is \(\frac{5x-9+5}{9}\), which simplifies to \(\frac{5x-4}{9}\).
03

Check if \(f\) and \(g\) are inverses

Two functions are inverses of each other if \(f(g(x)) = x\) and \(g(f(x)) = x\). By comparing, it can be seen that neither \(f(g(x))\) nor \(g(f(x))\) are equal to \(x\). Thus, \(f\) and \(g\) are not inverses of each other.

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