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
(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
Answers
GPT-4o mini
To find the inverse of the function \( f(x) = -4\sqrt{x} - 1 \), we'll follow these steps:
1. **Replace \( f(x) \) with \( y \)**:
\[
y = -4\sqrt{x} - 1
\]
2. **Solve for \( x \) in terms of \( y \)**:
\[
y + 1 = -4\sqrt{x}
\]
\[
\sqrt{x} = -\frac{y + 1}{4}
\]
Now, squaring both sides gives:
\[
x = \left(-\frac{y + 1}{4}\right)^2
\]
Simplifying further:
\[
x = \frac{(y + 1)^2}{16}
\]
3. **Replace \( y \) with \( x \) to express the inverse function**:
\[
f^{-1}(x) = \frac{(x + 1)^2}{16}
\]
Next, we need to find the appropriate restriction on \( x \):
- The original function \( f(x) \) has a domain of \( [0, \infty) \) because \( \sqrt{x} \) is defined for \( x \geq 0 \).
- The corresponding range of \( f(x) \) goes from \(-1\) downwards (as \( x \) increases, \( f(x) \) decreases).
Thus, the range of \( f(x) \) is \( (-\infty, -1] \), which means the inverse function should be valid for \( x \leq -1 \).
Therefore, the inverse function is:
\[
f^{-1}(x) = \frac{(x + 1)^2}{16}, \quad x \leq -1
\]
So the correct response is:
\[
\boxed{f^{-1}(x) = \frac{(x + 1)^2}{16}, \quad x \leq -1}
\]
1. **Replace \( f(x) \) with \( y \)**:
\[
y = -4\sqrt{x} - 1
\]
2. **Solve for \( x \) in terms of \( y \)**:
\[
y + 1 = -4\sqrt{x}
\]
\[
\sqrt{x} = -\frac{y + 1}{4}
\]
Now, squaring both sides gives:
\[
x = \left(-\frac{y + 1}{4}\right)^2
\]
Simplifying further:
\[
x = \frac{(y + 1)^2}{16}
\]
3. **Replace \( y \) with \( x \) to express the inverse function**:
\[
f^{-1}(x) = \frac{(x + 1)^2}{16}
\]
Next, we need to find the appropriate restriction on \( x \):
- The original function \( f(x) \) has a domain of \( [0, \infty) \) because \( \sqrt{x} \) is defined for \( x \geq 0 \).
- The corresponding range of \( f(x) \) goes from \(-1\) downwards (as \( x \) increases, \( f(x) \) decreases).
Thus, the range of \( f(x) \) is \( (-\infty, -1] \), which means the inverse function should be valid for \( x \leq -1 \).
Therefore, the inverse function is:
\[
f^{-1}(x) = \frac{(x + 1)^2}{16}, \quad x \leq -1
\]
So the correct response is:
\[
\boxed{f^{-1}(x) = \frac{(x + 1)^2}{16}, \quad x \leq -1}
\]