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Chapter 1: Problem 59

Determine whether the function has an inverse function. If it does, then find the inverse function. $$g(x)=\frac{x}{8}$$

Short Answer

Expert verified
Yes, the function \(g(x) = \frac{x}{8}\) has an inverse function. The inverse function is \(g^{-1}(x) = 8x\).

Step by step solution

01

Check if the function is one-to-one

A function is one-to-one if it never takes on the same value twice. The function \(g(x) = \frac{x}{8}\) is a straight line with a positive slope, which means it is one-to-one. Therefore, an inverse function exists.
02

Replace the function with 'y'

To find the inverse of the function, first replace the function notation with 'y'. It becomes \(y = \frac{x}{8}\)
03

Swap 'x' and 'y'

The next step is to interchange 'x' and 'y'. This gives: \(x = \frac{y}{8}\)
04

Solve for 'y'

To get 'y' alone on one side, multiply both sides by 8: \(y = 8x\). This is the inverse function.
05

Replace 'y' with the inverse function notation

The final step is to replace 'y' with the notation for the inverse function, \(g^{-1}(x)\). This gives the inverse function: \(g^{-1}(x) = 8x\)

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Most popular questions from this chapter

Evaluate the function for the indicated values. \(h(x)=[x+3]\) (a) \(h(-2)\) (b) \(h\left(\frac{1}{2}\right)\) (c) \(h(4.2)\) (d) \(h(-21.6)\)

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