Question
Find the inverse function of \( f(x)=\sqrt{7 x+8} \) \( f^{-1}(x)=\square \)
Ask by Henry Mullins. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The inverse function is \( f^{-1}(x) = \frac{x^2 - 8}{7} \).
Solution
1. Start with the function:
\[
f(x)=\sqrt{7x+8}
\]
2. Write the equation using \( y \):
\[
y=\sqrt{7x+8}
\]
3. Swap \( x \) and \( y \) to find the inverse:
\[
x=\sqrt{7y+8}
\]
4. Square both sides to eliminate the square root:
\[
x^2 = 7y+8
\]
5. Solve for \( y \):
\[
7y = x^2 - 8
\]
\[
y=\frac{x^2-8}{7}
\]
6. Thus, the inverse function is:
\[
f^{-1}(x)=\frac{x^2-8}{7}
\]
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Simplify this solution The Deep Dive
To find the inverse function, start by replacing \( f(x) \) with \( y \): \[ y = \sqrt{7x + 8} \] Next, solve for \( x \) in terms of \( y \). First, square both sides to eliminate the square root: \[ y^2 = 7x + 8 \] Now, isolate \( x \): \[ 7x = y^2 - 8 \] \[ x = \frac{y^2 - 8}{7} \] Now, replace \( y \) with \( x \) to express the inverse function: \[ f^{-1}(x) = \frac{x^2 - 8}{7} \] So, the inverse function is: \[ f^{-1}(x) = \frac{x^2 - 8}{7} \]
