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Chapter 8: Problem 20

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=3 x-7 \text { and } g(x)=\frac{x+3}{7}$$

Short Answer

Expert verified
The composite function \(f(g(x)) = \frac{3x-40}{7}\) and \(g(f(x)) = \frac{3x - 4}{7}\), and hence functions \(f\) and \(g\) are not inverses of each other.

Step by step solution

01

Compute \(f(g(x))\)

To find \(f(g(x))\), one must substitute \(g(x)\) into \(f(x)\). So \(f(g(x)) = 3 \cdot g(x) - 7 = 3 \cdot (\frac{x+3}{7}) - 7\). Now, simplify:
02

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

Multiplying each term by 3 and subtract 7: \(f(g(x)) = 3 \cdot (\frac{x+3}{7}) - 7 = \frac{3x+9}{7} - 7 = \frac{3x+9 - 49}{7} = \frac{3x-40}{7}\.
03

Compute \(g(f(x))\)

To find \(g(f(x))\), substitute \(f(x)\) into \(g(x)\). So \(g(f(x)) = \frac{f(x) + 3}{7} = \frac{(3x - 7) + 3}{7}\). Simplify this expression next.
04

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

Simplify by expanding inside the parenthesis to receive \(g(f(x)) = \frac{3x - 4}{7}\).
05

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

Two functions are inverses if and only if both \(f(g(x)) = x\) and \(g(f(x)) = x\). For these given functions \(f(g(x)) = \frac{3x-40}{7}\) and \(g(f(x)) = \frac{3x - 4}{7}\), neither of these equally \(x\).

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Most popular questions from this chapter

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

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$$\text { Simplify: }\left(\frac{3 x^{2} y^{-2}}{y^{3}}\right)^{-2} \cdot \text { (Section 5.7, Example 6) }$$

Show that $$f(x)=\frac{3 x-2}{5 x-3}$$ is its own inverse.

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