To find the inverse of the function [tex]\( f(x) = 2 - 2x \)[/tex], we need to follow these steps:
1. Write down the equation for [tex]\( f(x) \)[/tex]:
[tex]\[
y = 2 - 2x
\][/tex]
2. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to solve for the inverse function:
[tex]\[
x = 2 - 2y
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
- First, isolate the term involving [tex]\( y \)[/tex] by subtracting 2 from both sides:
[tex]\[
x - 2 = -2y
\][/tex]
- Next, divide both sides by -2 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{2 - x}{2}
\][/tex]
4. Simplify the equation:
[tex]\[
y = 1 - \frac{x}{2}
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = 1 - \frac{x}{2}
\][/tex]
Now, let's compare this with the given options:
1. [tex]\( f^{-1}(x) = \frac{4}{5} \)[/tex]
2. [tex]\( f^{-1}(x) = -\frac{1}{2}x + 1 \)[/tex]
3. [tex]\( f^{-1}(x) = -\frac{1}{4}x + \frac{4}{5} \)[/tex]
4. [tex]\( f^{-1}(x) = \frac{1}{2}x - 1 \)[/tex]
We see that none of the options exactly match [tex]\( f^{-1}(x) = 1 - \frac{x}{2} \)[/tex]. However, if we rearrange this into a different form, we get:
[tex]\[
f^{-1}(x) = \frac{1} - \frac{x}{2}
\][/tex]
which simplifies to:
[tex]\[
f^{-1}(x) =\frac{1}{2} x-1
\][/tex]
Therefore, the correct inverse function among the given options is:
[tex]\[
f^{-1}(x) = \frac{1}{2} x - 1
\][/tex]
Hence, the correct choice is:
[tex]\[
\boxed{4}
\][/tex]
