To find the inverse of the function [tex]\( f(x) = 2 - 2x \)[/tex], let's go through the process step by step.
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 2 - 2x \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[ x = 2 - 2y \][/tex]
3. Solve the equation above for [tex]\( y \)[/tex]:
[tex]\[ x = 2 - 2y \][/tex]
[tex]\[ x - 2 = -2y \][/tex]
[tex]\[ y = \frac{2 - x}{2} \][/tex]
4. Simplify the expression:
[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 result with the given options:
- [tex]\( f^{-1}(x) = -\frac{1}{4}x + \frac{4}{5} \)[/tex]
- [tex]\( f^{-1}(x) = -\frac{1}{2}x + 1 \)[/tex]
- [tex]\( f^{-1}(x) = \frac{1}{2}x - 1 \)[/tex]
- [tex]\( f^{-1}(x) = \frac{4}{5} \)[/tex]
The correct function that matches our result [tex]\( f^{-1}(x) = 1 - \frac{x}{2} \)[/tex] is:
[tex]\[ f^{-1}(x) = -\frac{1}{2} x + 1 \][/tex]
Therefore, the correct answer is:
[tex]\[ f^{-1}(x) = -\frac{1}{2} x + 1 \][/tex]
