Question

find the inverse function in slope - intercept form (mx + b):
$f(x)=-x - 3$

Explanation:

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

\( y = -x - 3 \)

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

\( x = -y - 3 \)

Step3: Solve for \( y \)

Add \( y \) to both sides: \( x + y = -3 \)
Subtract \( x \) from both sides: \( y = -x - 3 \) Wait, no, let's do it correctly. Starting from \( x = -y - 3 \), add 3 to both sides: \( x + 3 = -y \), then multiply both sides by -1: \( y = -x - 3 \)? No, that's wrong. Wait, \( x = -y - 3 \), so to solve for \( y \), we can add \( y \) to both sides: \( x + y = -3 \), then subtract \( x \): \( y = -x - 3 \)? No, that's the original function. Wait, no, let's do it step by step.

Starting over: \( y = -x - 3 \)

Swap \( x \) and \( y \): \( x = -y - 3 \)

Now, solve for \( y \):

Add 3 to both sides: \( x + 3 = -y \)

Multiply both sides by -1: \( -x - 3 = y \), so \( y = -x - 3 \)? Wait, that can't be. Wait, no, the original function is \( f(x) = -x - 3 \), which is a linear function with slope -1. The inverse of a linear function \( y = mx + b \) is found by swapping \( x \) and \( y \) and solving for \( y \).

So \( x = -y - 3 \)

Let's solve for \( y \):

Add \( y \) to both sides: \( x + y = -3 \)

Subtract \( x \) from both sides: \( y = -x - 3 \). Wait, that's the same as the original function? But that would mean the function is its own inverse. Let's check: if \( f(x) = -x - 3 \), then \( f(f(x)) = f(-x - 3) = -(-x - 3) - 3 = x + 3 - 3 = x \), so yes, it is its own inverse. So the inverse function \( f^{-1}(x) = -x - 3 \)? Wait, no, wait, let's plug in a value. Let's take \( x = 0 \), then \( f(0) = -3 \). Now, the inverse function should take -3 back to 0. Let's plug -3 into \( f^{-1}(x) \): \( f^{-1}(-3) = -(-3) - 3 = 3 - 3 = 0 \), which works. Let's take another value: \( x = 1 \), \( f(1) = -1 - 3 = -4 \). Then \( f^{-1}(-4) = -(-4) - 3 = 4 - 3 = 1 \), which is correct. So yes, the inverse function is \( y = -x - 3 \), so in slope-intercept form, it's \( f^{-1}(x) = -x - 3 \). Wait, but that seems odd, but mathematically, it checks out because \( f(f(x)) = x \).

Wait, maybe I made a mistake in the solving. Let's do it again:

Original function: \( y = -x - 3 \)

Swap \( x \) and \( y \): \( x = -y - 3 \)

Solve for \( y \):

Add 3 to both sides: \( x + 3 = -y \)

Multiply both sides by -1: \( y = -x - 3 \). Yes, that's correct. So the inverse function is \( f^{-1}(x) = -x - 3 \).

Answer:

The inverse function in slope-intercept form is \( f^{-1}(x) = -x - 3 \)