To find the inverse of the function [tex]\( f(x) = \frac{1}{3} - \frac{1}{21} x \)[/tex], follow these steps:
1. Express the function as [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{3} - \frac{1}{21} x
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \frac{1}{3} - \frac{1}{21} y
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[
x = \frac{1}{3} - \frac{1}{21} y
\][/tex]
Isolate [tex]\( y \)[/tex] by first getting rid of the constant term on the right side:
[tex]\[
x - \frac{1}{3} = -\frac{1}{21} y
\][/tex]
Multiply both sides by [tex]\( -21 \)[/tex] to isolate [tex]\( y \)[/tex]:
[tex]\[
-21(x - \frac{1}{3}) = y
\][/tex]
Distribute [tex]\( -21 \)[/tex]:
[tex]\[
y = -21x + 7
\][/tex]
4. Write the inverse function:
[tex]\[
f^{-1}(x) = 7 - 21x
\][/tex]
So, the correct answer is:
D. [tex]\( f^{-1}(x) = 7 - 21x \)[/tex]
