To find the inverse of the given function \( f(x) = \frac{1}{3} - \frac{1}{21} x \), we must follow these steps:
1. Rewrite the function: Start with \( y = f(x) = \frac{1}{3} - \frac{1}{21} x \).
2. Swap variables: Let \( y \) and \( x \) switch places. This is because we are looking for the inverse function, \( x = f^{-1}(y) \). So we get:
[tex]\[ x = \frac{1}{3} - \frac{1}{21} y \][/tex]
3. Solve for \( y \): Next we need to isolate \( y \) in terms of \( x \):
- Start by subtracting \(\frac{1}{3}\) from both sides:
[tex]\[ x - \frac{1}{3} = -\frac{1}{21} y \][/tex]
- Multiply both sides by \(-21\) to isolate \( y \):
[tex]\[ -21 \left( x - \frac{1}{3} \right) = y \][/tex]
- Simplify the right-hand side:
[tex]\[ y = -21x + 7 \][/tex]
So the inverse function is:
[tex]\[ f^{-1}(x) = 7 - 21x \][/tex]
Therefore, the correct answer is B. [tex]\(f^{-1}(x) = 7 - 21x\)[/tex].
