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Chapter 1: Problem 62

Determine whether the function has an inverse function. If it does, then find the inverse function. $$f(x)=\frac{3 x+4}{5}$$

Short Answer

Expert verified
Yes, the function \(f(x)=\frac{3x+4}{5}\) does have an inverse and it is \(f^{-1}(x)=\frac{5x-4}{3}\).

Step by step solution

01

Switch 'y' and 'x'

Replace \(f(x)\) with 'y', then switch 'y' and 'x' in the equation. So, it will become \(x=\frac{3y+4}{5}\).
02

Solve for 'y'

Solve for 'y' by multiplying both sides by 5 to get \(5x=3y+4\). Then, subtract 4 from both sides, resulting in \(3y=5x-4\). Finally, divide all terms by 3 to solve for 'y'. \(y=\frac{5x-4}{3}\).
03

Verification

The inverse function, often denoted as \(f^{-1}(x)\) is thus \(f^{-1}(x)=\frac{5x-4}{3}\). The function and its inverse are both linear functions, and their slopes are negative reciprocals of each other, which implies that they are indeed inverses.

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Most popular questions from this chapter

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$g(x)=|x|-5$$

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