Question
Find the inverse of the function \(x=\sqrt{\frac{y}{2}+13}\)
Answers
GPT-4o mini
To find the inverse of the function given by
\[
x = \sqrt{\frac{y}{2} + 13},
\]
we start by solving for \(y\) in terms of \(x\).
1. First, square both sides to eliminate the square root:
\[
x^2 = \frac{y}{2} + 13.
\]
2. Next, isolate \(\frac{y}{2}\) by subtracting 13 from both sides:
\[
x^2 - 13 = \frac{y}{2}.
\]
3. Now, multiply both sides by 2 to solve for \(y\):
\[
y = 2(x^2 - 13).
\]
4. Simplifying the equation gives:
\[
y = 2x^2 - 26.
\]
Now that we have expressed \(y\) in terms of \(x\), we can write the inverse function. The inverse of the function \(y\) would be:
\[
f^{-1}(x) = 2x^2 - 26.
\]
Thus, the inverse of the function \(x = \sqrt{\frac{y}{2} + 13}\) is
\[
f^{-1}(x) = 2x^2 - 26.
\]
\[
x = \sqrt{\frac{y}{2} + 13},
\]
we start by solving for \(y\) in terms of \(x\).
1. First, square both sides to eliminate the square root:
\[
x^2 = \frac{y}{2} + 13.
\]
2. Next, isolate \(\frac{y}{2}\) by subtracting 13 from both sides:
\[
x^2 - 13 = \frac{y}{2}.
\]
3. Now, multiply both sides by 2 to solve for \(y\):
\[
y = 2(x^2 - 13).
\]
4. Simplifying the equation gives:
\[
y = 2x^2 - 26.
\]
Now that we have expressed \(y\) in terms of \(x\), we can write the inverse function. The inverse of the function \(y\) would be:
\[
f^{-1}(x) = 2x^2 - 26.
\]
Thus, the inverse of the function \(x = \sqrt{\frac{y}{2} + 13}\) is
\[
f^{-1}(x) = 2x^2 - 26.
\]