To find the inverse of the function [tex]\( f(x) = \frac{3 - x}{7} \)[/tex], follow these steps:
1. Start with the given function:
[tex]\[ f(x) = \frac{3 - x}{7} \][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex] for simplicity:
[tex]\[ y = \frac{3 - x}{7} \][/tex]
3. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to set up for solving the inverse:
[tex]\[ x = \frac{3 - y}{7} \][/tex]
4. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
- First, multiply both sides by 7 to clear the fraction:
[tex]\[ 7x = 3 - y \][/tex]
- Next, isolate [tex]\( y \)[/tex] by subtracting 3 from both sides:
[tex]\[ 7x - 3 = -y \][/tex]
- Then, multiply both sides by -1:
[tex]\[ y = 3 - 7x \][/tex]
5. State the inverse function:
[tex]\[ f^{-1}(x) = 3 - 7x \][/tex]
So, the correct inverse function is:
[tex]\[ f^{-1}(x) = 3 - 7x \][/tex]
Thus, the correct answer is Option D.
