To find the inverse of the function \( f(x) = -4\sqrt{x} - 1 \), we start by replacing \( f(x) \) with \( y \):
\[ y = -4\sqrt{x} - 1 \]
Next, we solve for \( x \) in terms of \( y \).
-
Add 1 to both sides: \[ y + 1 = -4\sqrt{x} \]
-
Divide by -4: \[ \sqrt{x} = -\frac{y + 1}{4} \]
-
Square both sides: \[ x = \left(-\frac{y + 1}{4}\right)^2 \] \[ x = \frac{(y + 1)^2}{16} \]
Now, swap \( x \) and \( y \) to express the inverse function: \[ f^{-1}(x) = \frac{(x + 1)^2}{16} \]
Next, we must determine the correct domain for the inverse function. The original function \( f(x) = -4\sqrt{x} - 1 \) has the domain \( x \geq 0 \). The function is decreasing, leading to its range being \( (-\infty, -1] \). Thus, the inverse function will have a domain of \( x \leq -1 \) since the output of \( f(x) \) lies in that range.
Therefore, the inverse function can be written as:
\[ f^{-1}(x) = \frac{(x + 1)^2}{16}, \quad x \leq -1 \]
Finally, the correct response is:
\( f^{-1}(x) = \frac{(x + 1)^2}{16}, \quad x \leq -1 \)
1 answer