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?

Question

(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

Answer 1

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

\[ y = (x - 1)^2 - 4 \]

Next, solve for \( x \) in terms of \( y \):

1. Add 4 to both sides:

\[ y + 4 = (x - 1)^2 \]

2. Take the square root of both sides:

\[ \sqrt{y + 4} = |x - 1| \]

Since we have restricted the domain to \( x \leq 1 \), we discard the positive root:

\[ x - 1 = -\sqrt{y + 4} \]

3. Solve for \( x \):

\[ x = 1 - \sqrt{y + 4} \]

Now, substituting \( y \) back with \( x \) for the inverse function:

\[ f^{-1}(x) = 1 - \sqrt{x + 4} \]

Now we determine the domain of the inverse. The output of \( f(x) \) when \( x \leq 1 \) can be found by evaluating \( f(1) \):

\[ f(1) = (1 - 1)^2 - 4 = -4 \]

As \( x \) decreases from 1, the function value increases towards positive infinity. Thus, the range of \( f(x) \) (and consequently the domain of \( f^{-1}(x) \)) is:

\[ y \geq -4 \]

Combining this information, the inverse function is:

\[ f^{-1}(x) = 1 - \sqrt{x + 4}, \quad x \geq -4 \]

Thus, the correct response is:

\[ f^{-1}(x) = 1 - \sqrt{x + 4}, \quad x \geq -4 \]