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Chapter 1: Problem 60

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

Short Answer

Expert verified
Yes, the function \(f(x)=3x+5\) has an inverse. The inverse function is \(f^{-1}(x)=(x-5)/3\).

Step by step solution

01

Verify the function is one-to-one

A function is one-to-one if it passes both the vertical line test and horizontal line test. The given function \(f(x)=3x+5\) is linear and passes both tests, meaning it is a one-to-one function. Hence, an inverse function exists.
02

Rewrite the function equation in terms of y

Replace \(f(x)\) with \(y\). This gives us the equation \(y=3x+5\).
03

Interchange x and y

Switch the roles of x and y in the equation to get \(x=3y+5\). The new equation represents the inverse of the original function.
04

Solve for y

To isolate y, subtract 5 from both sides to get \(x-5=3y\), and then divide by 3. The result is \(y=(x-5)/3\).
05

Write the inverse function

Replace \(y\) with \(f^{-1}(x)\) to get the inverse function \(f^{-1}(x)=(x-5)/3\).

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