Question

The function lower f left-parenthesis x right-parenthesis equals left-parenthesis x minus 1 right-parenthesis squared minus 4 is not one-to-one. If you restrict the domain for f left-parenthesis x right-parenthesis to x less-than-or-equal-to 1, what is its inverse function and the domain for the inverse?
(1 point)
Responses

lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 minus Start Root x plus 4 End Root, x greater-than-or-equal-to negative 4
Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 minus Start Root x plus 4 End Root, x greater-than-or-equal-to negative 4

lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 plus Start Root x plus 4 End Root, x greater-than-or-equal-to negative 4
Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 plus Start Root x plus 4 End Root, x greater-than-or-equal-to negative 4

lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 minus Start Root x minus 4 End Root, x greater-than-or-equal-to 4
Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 minus Start Root x minus 4 End Root, x greater-than-or-equal-to 4

lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 plus Start Root x minus 4 End Root, x greater-than-or-equal-to 4
Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 plus Start Root x minus 4 End Root, x greater-than-or-equal-to 4

Answer 1

To find the inverse of the function \( f(x) = (x - 1)^2 - 4 \) with the domain restricted to \( x \leq 1 \), we first need to express this function in terms of \( y \):

1. Start with:
\[
y = (x - 1)^2 - 4
\]

2. To find the inverse, we solve for \( x \) in terms of \( y \):
\[
y + 4 = (x - 1)^2
\]
\[
\sqrt{y + 4} = |x - 1|
\]

Since we are considering the domain \( x \leq 1 \), this means we only consider the negative part:
\[
x - 1 = -\sqrt{y + 4}
\]
\[
x = 1 - \sqrt{y + 4}
\]

3. Now we can express the inverse function:
\[
f^{-1}(y) = 1 - \sqrt{y + 4}
\]

Next, we need to determine the appropriate domain for the inverse function. The original function \( f(x) \) at its maximum occurs when \( x = 1 \):
\[
f(1) = (1 - 1)^2 - 4 = -4
\]

As \( x \) decreases from 1, \( f(x) \) increases and can take on all values greater than or equal to \(-4\). Hence, the range of \( f \) when \( x \leq 1 \) becomes the domain of \( f^{-1} \):

- Therefore, the domain of \( f^{-1}(x) \) is:
\[
x \geq -4
\]

Putting it all together, we have the inverse function as:
\[
f^{-1}(x) = 1 - \sqrt{x + 4}, \quad x \geq -4
\]

The correct response is:
\[
f^{-1}(x) = 1 - \sqrt{x + 4}, \quad x \geq -4
\]