Question

1. find the inverse for the following function. (4pts) f(x) = -2x - 10

Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start by writing the function as \( y = -2x - 10 \). This is a standard first step in finding the inverse of a function, as it makes it easier to swap the variables \( x \) and \( y \) later.

Step2: Swap \( x \) and \( y \)

After swapping, we get \( x = -2y - 10 \). Swapping the variables is based on the definition of an inverse function, where if \( y = f(x) \), then \( x = f^{-1}(y) \), and we want to solve for \( y \) in terms of \( x \) to get the inverse function.

Step3: Solve for \( y \)

First, add 10 to both sides of the equation: \( x + 10 = -2y \). Then, divide both sides by -2: \( y=\frac{x + 10}{-2}=-\frac{1}{2}x - 5 \).

Step4: Replace \( y \) with \( f^{-1}(x) \)

The inverse function is \( f^{-1}(x)=-\frac{1}{2}x - 5 \). This notation is used to represent the inverse of the function \( f(x) \).

Answer:

The inverse function of \( f(x)=-2x - 10 \) is \( f^{-1}(x)=-\frac{1}{2}x - 5 \)