Question
Find the inverse function. \( f(x)=x^{2}+22 \) \( \left.f^{-1}(x)=\sqrt[{[?}]\right]{x+\square} \)
Ask by Mcdonald Vaughn. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The inverse function is \( f^{-1}(x) = \sqrt{x - 22} \).
Solution
To find the inverse of the function \( f(x) = x^2 + 22 \), follow these steps:
1. **Replace \( f(x) \) with \( y \):**
\[
y = x^2 + 22
\]
2. **Solve for \( x \) in terms of \( y \):**
\[
y - 22 = x^2
\]
\[
x = \pm\sqrt{y - 22}
\]
3. **Define the inverse function:**
Since a function must pass the vertical line test and have only one output for each input, we typically take the positive square root when defining the inverse (assuming the original function \( f(x) \) is restricted to \( x \geq 0 \)).
\[
f^{-1}(x) = \sqrt{x - 22}
\]
So, the inverse function is:
\[
f^{-1}(x) = \sqrt{x - 22}
\]
**Final Answer:**
\( f^{-1}(x) = \sqrt{x\ -\ 22} \)
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Simplify this solution Bonus Knowledge
To find the inverse function of \( f(x) = x^2 + 22 \), we start by switching \( f(x) \) and \( x \): 1. Set \( y = x^2 + 22 \). 2. To find the inverse, express \( x \) in terms of \( y \): \[ y - 22 = x^2 \] \[ x = \sqrt{y - 22} \] (considering \( x \geq 0 \) because it's a function of \( x^2 \)). So, the inverse function is \( f^{-1}(x) = \sqrt{x - 22} \). Now filling in the blanks: \[ \left.f^{-1}(x)=\sqrt[{2}]{x-22}\right. \] If you need more expression options, let me know!
