Question: The function $f(x) = x + 14$ 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 $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
a. Select the correct choice below and fill in the answer box(es) to complete your choice.
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
A. $f^{-1}(x) = __$, for all $x$
B. $f^{-1}(x) = __$, for $x \leq __$
C. $f^{-1}(x) = __$, for $x \geq __$
D. $f^{-1}(x) = __$, for $x \neq __$
b. Verify that the equation is correct.
$f(f^{-1}(x)) = f(__)$ and $f^{-1}(f(x)) = f^{-1}(__)$ Substitute.
$= __$ $= __$ Simplify.
The equation is __
1. **State the problem:** We are given the function $f(x) = x + 14$ and asked to find its inverse function $f^{-1}(x)$.
2. **Find the inverse function:**
To find $f^{-1}(x)$, start with $y = f(x) = x + 14$.
Swap $x$ and $y$ to get $x = y + 14$.
Solve for $y$:
$$y = x - 14$$
So, the inverse function is:
$$f^{-1}(x) = x - 14$$
Since $f$ is a linear function with no domain restrictions, the inverse is valid for all real $x$.
3. **Select the correct choice:**
Choice A: $f^{-1}(x) = x - 14$, for all $x$.
4. **Verify the inverse function:**
- Compute $f(f^{-1}(x))$:
$$f(f^{-1}(x)) = f(x - 14) = (x - 14) + 14 = x$$
- Compute $f^{-1}(f(x))$:
$$f^{-1}(f(x)) = f^{-1}(x + 14) = (x + 14) - 14 = x$$
5. **Conclusion:**
Both compositions return $x$, confirming that $f^{-1}(x) = x - 14$ is the correct inverse function.
**Final answers:**
- $f^{-1}(x) = x - 14$, for all $x$.
- Verification:
$$f(f^{-1}(x)) = x$$
$$f^{-1}(f(x)) = x$$
The equation is correct.