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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
So the composites are \(f(g(x)) = \frac{5x - 56}{9}\) and \(g(f(x)) = \frac{5x-4}{9}\). The functions \(f(x) = 5x - 9\) and \(g(x) = \frac{x+5}{9}\) are not inverses of each other.

Step by step solution

01

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

Substitute \(g(x)\) into \(f(x)\). This gives us, \(f(g(x)) = f(\frac{x+5}{9}) = 5(\frac{x+5}{9}) - 9 = \frac{5x+25}{9} - 9 = \frac{5x+25}{9} - \frac{81}{9}\). Simplifying yields: \(f(g(x)) = \frac{5x - 56}{9}\)
02

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

Now replace \(f(x)\) in \(g(x)\), You will get, \(g(f(x)) = g(5x-9) = \frac{(5x-9)+5}{9} = \frac{5x-9+5}{9} = \frac{5x-4}{9}\)
03

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

The functions are inverses of each other if and only if both \(f(g(x)) = x\) and \(g(f(x)) = x\). In this case, \(f(g(x))\) simplifies to \(\frac{5x - 56}{9}\) and \(g(f(x))\) simplifies to \(\frac{5x-4}{9}\), and neither of them equals \(x\). So the functions \(f(x) = 5x - 9\) and \(g(x) = \frac{x+5}{9}\) are not inverses of each other.

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