To determine the inverse of the function [tex]\( f(x) = \frac{3 - x}{7} \)[/tex], we need to follow a series of steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = \frac{3 - x}{7}
\][/tex]
2. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This step helps us find the inverse function:
[tex]\[
x = \frac{3 - y}{7}
\][/tex]
3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
- First, clear the fraction by multiplying both sides by 7:
[tex]\[
7x = 3 - y
\][/tex]
- Next, isolate [tex]\( y \)[/tex] by subtracting 3 from both sides:
[tex]\[
7x - 3 = -y
\][/tex]
- Finally, multiply both sides by -1 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 3 - 7x
\][/tex]
4. Replace [tex]\( y \)[/tex] with [tex]\( f^{-1}(x) \)[/tex] to express the inverse function:
[tex]\[
f^{-1}(x) = 3 - 7x
\][/tex]
After performing all the steps, we determine that the inverse of the function [tex]\( f(x) = \frac{3 - x}{7} \)[/tex] is:
[tex]\[
f^{-1}(x) = 3 - 7x
\][/tex]
The correct answer is:
A. [tex]\( f^{-1}(x) = 3 - 7x \)[/tex]
