The function $$ f(x)=\frac{21}{x} $$ is one-to-one. a. Find an equation for $$ f^{-1}(x) $$, the inverse function. b. Verify that your equation is correct by showing that $$ f\left(f^{-1}(x)\right)=x $$ and $$ f^{-1}(f(x))=x $$. A. $$ f^{-1}(x)=\frac{21}{x} $$, for $$ x \neq 0 $$ B. $$ f^{-1}(x)=\square $$, for $$ x \geq \square $$ C. $$ f^{-1}(x)=\quad $$, for all $$ x $$ D. $$ f^{-1}(x)=\square $$, for $$ x \leq \square $$ b. Verify that the equation is correct. $$ \begin{array}{rlrlrl} \mathrm{f}\left(\mathrm{f}^{-1}(\mathrm{x})\right) & =\mathrm{f}(\square) & \text { and } & \mathrm{f}^{-1}(\mathrm{f}(\mathrm{x})) & =\mathrm{f}^{-1}(\square) & \ & =\square & & \text { Substitute. } \ & =\square & & \text { Simplify } . \end{array} $$

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To solve the problem, we will follow the steps outlined:

### Part a: Find the Inverse Function

1. **Start with the function**:
   $$
   f(x) = \frac{21}{x}
   $$

2. **Replace $$ f(x) $$ with $$ y $$**:
   $$
   y = \frac{21}{x}
   $$

3. **Swap $$ x $$ and $$ y $$** to find the inverse:
   $$
   x = \frac{21}{y}
   $$

4. **Solve for $$ y $$**:
   - Multiply both sides by $$ y $$:
   $$
   xy = 21
   $$
   - Divide both sides by $$ x $$:
   $$
   y = \frac{21}{x}
   $$

5. **Thus, the inverse function is**:
   $$
   f^{-1}(x) = \frac{21}{x}
   $$

### Part b: Verify the Inverse Function

We need to verify that:
1. $$ f(f^{-1}(x)) = x $$
2. $$ f^{-1}(f(x)) = x $$

#### Verification of $$ f(f^{-1}(x)) $$

1. **Substitute $$ f^{-1}(x) $$ into $$ f(x) $$**:
   $$
   f(f^{-1}(x)) = f\left(\frac{21}{x}\right)
   $$

2. **Calculate $$ f\left(\frac{21}{x}\right) $$**:
   $$
   f\left(\frac{21}{x}\right) = \frac{21}{\frac{21}{x}} = \frac{21 \cdot x}{21} = x
   $$

#### Verification of $$ f^{-1}(f(x)) $$

1. **Substitute $$ f(x) $$ into $$ f^{-1}(x) $$**:
   $$
   f^{-1}(f(x)) = f^{-1}\left(\frac{21}{x}\right)
   $$

2. **Calculate $$ f^{-1}\left(\frac{21}{x}\right) $$**:
   $$
   f^{-1}\left(\frac{21}{x}\right) = \frac{21}{\frac{21}{x}} = \frac{21 \cdot x}{21} = x
   $$

### Conclusion

Both verifications confirm that the inverse function is correct. Therefore, the answers are:

- **A**: $$ f^{-1}(x) = \frac{21}{x} $$, for $$ x \neq 0 $$
- **B**: $$ f^{-1}(x) = \frac{21}{x} $$, for $$ x \geq 0 $$
- **C**: $$ f^{-1}(x) = \frac{21}{x} $$, for all $$ x $$
- **D**: $$ f^{-1}(x) = \frac{21}{x} $$, for $$ x \leq 0 $$

And the verification results are:
$$
\begin{array}{rlrlrl}
f(f^{-1}(x)) & = f\left(\frac{21}{x}\right) & \text{ and } & f^{-1}(f(x)) & = f^{-1}\left(\frac{21}{x}\right) & \
& = x & & \text{ Substitute. } \
& = x & & \text{ Simplify. }
\end{array}
$$
The inverse function is $$ f^{-1}(x) = \frac{21}{x} $$ for $$ x \neq 0 $$. To verify: - $$ f(f^{-1}(x)) = x $$ - $$ f^{-1}(f(x)) = x $$ These confirm that the inverse is correct.

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