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Chapter 1: Problem 13

Verify that \(f\) and \(g\) are inverse functions. $$f(x)=-\frac{7}{2} x-3, \quad g(x)=-\frac{2 x+6}{7}$$

Short Answer

Expert verified
Yes, \( f(x) \) and \( g(x) \) are inverse functions.

Step by step solution

01

Substituting \( f(x) \) into \( g(x) \)

In the first step, substitute \( f(x) = -\frac{7}{2} x - 3 \) in place of \( x \) in the function \( g \). Thus, \( g(f(x)) = -\frac{2(-\frac{7}{2}x - 3) +6}{7} \).
02

Simplifying \( g(f(x)) \)

Now simplify the expression: \( g(f(x)) = -\frac{2(-\frac{7}{2}x - 3) +6}{7} = -\frac{-7x - 6 + 6}{7} = \frac{7x}{7} = x \). So \( g(f(x)) = x \), meaning that \( g \) is the inverse of \( f \).
03

Substituting \( g(x) \) into \( f(x) \)

Now, substitute \( g(x) = -\frac{2x + 6}{7} \) in place of \( x \) in the function \( f \). Thus, \( f(g(x)) = -\frac{7}{2}(-\frac{2x + 6}{7}) - 3 \).
04

Simplifying \( f(g(x)) \)

Next, simplify the expression \( f(g(x)) = -\frac{7}{2}(-\frac{2x + 6}{7}) - 3 = -(-x - 3) - 3 = x + 3 - 3 = x \). Thus, \( f(g(x)) = x \), which means that \( f \) is the inverse of \( g \).

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Most popular questions from this chapter

Wages A mechanic's pay is 14.00 dollars per hour for regular time and time- and-a-half for overtime. The weekly wage function is \(W(h)=\left\\{\begin{array}{ll}14 h, & 040\end{array}\right.\) where \(h\) is the number of hours worked in a week. (a) Evaluate \(W(30), W(40), W(45),\) and \(W(50)\) (b) The company increased the regular work week to 45 hours. What is the new weekly wage function?

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