To determine the inverse of the function [tex]\( f(x) = \sqrt{x - 1} \)[/tex], we need to follow these steps:
1. Define the original function: [tex]\( f(x) = \sqrt{x - 1} \)[/tex].
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]: [tex]\( y = \sqrt{x - 1} \)[/tex].
3. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]: To find the inverse function, interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \sqrt{y - 1}
\][/tex]
4. Solve for [tex]\( y \)[/tex]:
- Square both sides to eliminate the square root:
[tex]\[
x^2 = y - 1
\][/tex]
- Solve for [tex]\( y \)[/tex]:
[tex]\[
y = x^2 + 1
\][/tex]
Thus, the inverse function of [tex]\( f(x) = \sqrt{x - 1} \)[/tex] is [tex]\( f^{-1}(x) = x^2 + 1 \)[/tex].
Therefore, the correct answer is:
[tex]\[ f^{-1}(x) = x^2 + 1 \][/tex]
