To find the inverse of the function [tex]\( f(x) = (x-1)^2 - 4 \)[/tex] with the domain restriction [tex]\( x \leq 1 \)[/tex], we need to follow these steps:
### Step 1: Express [tex]\( f(x) \)[/tex] in terms of [tex]\( y \)[/tex]
Let [tex]\( y = f(x) \)[/tex], therefore:
[tex]\[ y = (x-1)^2 - 4 \][/tex]
### Step 2: Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]
To find the inverse function, we solve for [tex]\( x \)[/tex]:
[tex]\[ y + 4 = (x-1)^2 \][/tex]
[tex]\[ x - 1 = \pm \sqrt{y + 4} \][/tex]
Since we have restricted the domain to [tex]\( x \leq 1 \)[/tex], we choose the negative branch because the square root will produce two values and we need to select the value that satisfies [tex]\( x \leq 1 \)[/tex]:
[tex]\[ x - 1 = -\sqrt{y + 4} \][/tex]
[tex]\[ x = 1 - \sqrt{y + 4} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(y) = 1 - \sqrt{y + 4} \][/tex]
### Step 3: Determine the domain of the inverse function
The domain of the inverse function corresponds to the range of the original function [tex]\( f(x) \)[/tex] with the restricted domain [tex]\( x \leq 1 \)[/tex].
Analyzing [tex]\( f(x) = (x-1)^2 - 4 \)[/tex] with [tex]\( x \leq 1 \)[/tex]:
- As [tex]\( x \)[/tex] decreases from [tex]\( 1 \)[/tex] to [tex]\( - \infty \)[/tex], [tex]\((x-1)^2\)[/tex] increases from [tex]\( 0 \)[/tex] to [tex]\( + \infty \)[/tex].
- Thus, [tex]\( f(x) = (x-1)^2 - 4 \)[/tex] decreases from [tex]\( 0 - 4 = -4 \)[/tex] to [tex]\( + \infty - 4 = -4 + \infty = + \infty \)[/tex].
Therefore, the range of [tex]\( f(x) \)[/tex] when [tex]\( x \leq 1 \)[/tex] is [tex]\( (- \infty, -4] \)[/tex].
Thus, the domain for the inverse function [tex]\( f^{-1}(y) \)[/tex] is:
[tex]\[ y \leq -4 \][/tex]
### Summary
The inverse function of [tex]\( f(x) \)[/tex] when the domain is restricted to [tex]\( x \leq 1 \)[/tex] is:
[tex]\[ f^{-1}(y) = 1 - \sqrt{y + 4} \][/tex]
And the domain for [tex]\( f^{-1}(y) \)[/tex] is:
[tex]\[ y \leq -4 \][/tex]
