Answer :
To find the inverse of the function \( f(x) = \frac{1}{3} - \frac{1}{21} x \), we need to follow these steps:
1. Rewrite the function: Begin with the equation that defines \( f(x) \):
[tex]\[ y = \frac{1}{3} - \frac{1}{21} x \][/tex]
2. Swap \(x\) and \(y\): Interchange the roles of \(x\) and \(y\) because we are now solving for the inverse function:
[tex]\[ x = \frac{1}{3} - \frac{1}{21} y \][/tex]
3. Solve for \(y\): Isolate \(y\) to express it in terms of \(x\). Start by moving the constant on the right-hand side:
[tex]\[ x - \frac{1}{3} = - \frac{1}{21} y \][/tex]
[tex]\[ - \frac{1}{21} y = x - \frac{1}{3} \][/tex]
Now, multiply both sides by \(-21\) to solve for \(y\):
[tex]\[ y = -21 (x - \frac{1}{3}) \][/tex]
4. Simplify the expression for \(y\): Distribute the \(-21\):
[tex]\[ y = -21x + 7 \][/tex]
So, the inverse function is:
[tex]\[ f^{-1}(x) = 7 - 21x \][/tex]
5. Compare with the given choices: Let's identify if this matches any of the provided options:
A. \(f^{-1}(x) = 7 - 21x\)
B. \(f^{-1}(x) = \frac{1}{7} - \frac{1}{21} x\)
C. \(f^{-1}(x) = \frac{1}{7} - 21x\)
D. \(f^{-1}(x) = 7 - \frac{1}{21} x\)
Clearly, the correct answer is:
[tex]\[ \boxed{A. \; f^{-1}(x) = 7 - 21x} \][/tex]
1. Rewrite the function: Begin with the equation that defines \( f(x) \):
[tex]\[ y = \frac{1}{3} - \frac{1}{21} x \][/tex]
2. Swap \(x\) and \(y\): Interchange the roles of \(x\) and \(y\) because we are now solving for the inverse function:
[tex]\[ x = \frac{1}{3} - \frac{1}{21} y \][/tex]
3. Solve for \(y\): Isolate \(y\) to express it in terms of \(x\). Start by moving the constant on the right-hand side:
[tex]\[ x - \frac{1}{3} = - \frac{1}{21} y \][/tex]
[tex]\[ - \frac{1}{21} y = x - \frac{1}{3} \][/tex]
Now, multiply both sides by \(-21\) to solve for \(y\):
[tex]\[ y = -21 (x - \frac{1}{3}) \][/tex]
4. Simplify the expression for \(y\): Distribute the \(-21\):
[tex]\[ y = -21x + 7 \][/tex]
So, the inverse function is:
[tex]\[ f^{-1}(x) = 7 - 21x \][/tex]
5. Compare with the given choices: Let's identify if this matches any of the provided options:
A. \(f^{-1}(x) = 7 - 21x\)
B. \(f^{-1}(x) = \frac{1}{7} - \frac{1}{21} x\)
C. \(f^{-1}(x) = \frac{1}{7} - 21x\)
D. \(f^{-1}(x) = 7 - \frac{1}{21} x\)
Clearly, the correct answer is:
[tex]\[ \boxed{A. \; f^{-1}(x) = 7 - 21x} \][/tex]
