If f(x) = 8 / sqrt(x + 2), find a formula for f^(-1)(y) and find the domain for f^(-1).

Steps to Solve

1. Rewrite the function Start with the given function: $$ f(x) = \frac{8}{\sqrt{x + 2}} $$

2. Set up for the inverse function To find the inverse, replace $f(x)$ with $y$: $$ y = \frac{8}{\sqrt{x + 2}} $$

3. Solve for x in terms of y Multiply both sides by $\sqrt{x + 2}$: $$ y \sqrt{x + 2} = 8 $$ Now divide by $y$ (assuming $y \neq 0$): $$ \sqrt{x + 2} = \frac{8}{y} $$

4. Square both sides Square both sides to eliminate the square root: $$ x + 2 = \left(\frac{8}{y}\right)^2 $$ This simplifies to: $$ x + 2 = \frac{64}{y^2} $$

5. Isolate x Subtract 2 from both sides: $$ x = \frac{64}{y^2} - 2 $$

6. Write the inverse function The inverse function is: $$ f^{-1}(y) = \frac{64}{y^2} - 2 $$

7. Determine the domain of the inverse function The domain of the inverse function corresponds to the range of the original function. Since the output of $f(x)$ must be positive, we find the minimum value:

• As $x \to -2$, $f(x) \to 4$ (since $f(-2) = 4$)
• As $x \to \infty$, $f(x) \to 0$

Thus, the range of $f(x)$ is $(0, 4]$.

8. Write the domain of the inverse function Therefore, the domain of $f^{-1}(y)$ is: $$ 0 < y \leq 4 $$