Question

Question
For the function lower f left-parenthesis x right-parenthesis equals negative 4 Start Root x End Root minus 1, find the inverse function.
(1 point)
Responses

lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction, x less-than-or-equal-to negative 1
Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction, x less-than-or-equal-to negative 1

lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction, x greater-than-or-equal-to negative 1
Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction, x greater-than-or-equal-to negative 1

lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction
Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction

lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction, x less-than-or-equal-to negative 1
Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction, x less-than-or-equal-to negative 1
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Answer 1

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 \( y \).

1. Add 1 to both sides:
\[
y + 1 = -4\sqrt{x}
\]

2. Divide by -4:
\[
\sqrt{x} = -\frac{y + 1}{4}
\]

3. 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:

\( f^{-1}(x) = \frac{(x + 1)^2}{16}, \quad x \leq -1 \)