To find the inverse of the function [tex]\( f(x) = \frac{3 - x}{7} \)[/tex], we need to follow these steps:
1. Rewrite the function using a different variable, typically [tex]\( y \)[/tex]:
[tex]\[
y = \frac{3 - x}{7}
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. This involves rearranging the equation to isolate [tex]\( x \)[/tex].
Start by multiplying both sides by 7 to get rid of the fraction:
[tex]\[
7y = 3 - x
\][/tex]
Next, isolate [tex]\( x \)[/tex] by subtracting 3 from both sides:
[tex]\[
7y - 3 = -x
\][/tex]
Multiply both sides by -1 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 3 - 7y
\][/tex]
3. Interchange the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to write the inverse function. This means replacing [tex]\( y \)[/tex] with [tex]\( x \)[/tex] in the final expression from step 2:
[tex]\[
f^{-1}(x) = 3 - 7x
\][/tex]
Therefore, 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]
