To determine the inverse of the function [tex]\( f(x) = -5x - 4 \)[/tex], we need to follow several steps to solve for the inverse function. Let's do this in a detailed, step-by-step manner.
1. Start with the original function [tex]\( y = -5x - 4 \)[/tex]:
[tex]\[
y = -5x - 4
\][/tex]
2. Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to begin solving for the inverse:
[tex]\[
x = -5y - 4
\][/tex]
3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
- First, add 4 to both sides to begin isolating [tex]\( y \)[/tex]:
[tex]\[
x + 4 = -5y
\][/tex]
- Next, divide both sides by [tex]\(-5\)[/tex]:
[tex]\[
y = \frac{-(x + 4)}{5}
\][/tex]
4. Simplify the expression for [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{x + 4}{5}
\][/tex]
[tex]\[
y = -\frac{1}{5}x - \frac{4}{5}
\][/tex]
So, the inverse function is:
[tex]\[
f^{-1}(x) = -\frac{1}{5}x - \frac{4}{5}
\][/tex]
Now we need to match this result with one of the given choices:
1. [tex]\( f^{-1}(x) = -\frac{1}{5}x - \frac{4}{5} \)[/tex]
2. [tex]\( f^{-1}(x) = -\frac{1}{5}x + \frac{4}{5} \)[/tex]
3. [tex]\( f^{-1}(x) = -4x + 5 \)[/tex]
4. [tex]\( f^{-1}(x) = 4x + 4 \)[/tex]
Clearly, the correct option is:
[tex]\[
f^{-1}(x) = -\frac{1}{5}x - \frac{4}{5}
\][/tex]
Therefore, the inverse function of [tex]\( f(x) = -5x - 4 \)[/tex] is [tex]\( -\frac{1}{5}x - \frac{4}{5} \)[/tex], which corresponds to:
[tex]\[
\boxed{1}
\][/tex]
