To find the inverse of the function [tex]\( f(x) = \sqrt{x} + 7 \)[/tex], we need to follow these steps:
1. Set up the equation: Start by setting [tex]\( y = f(x) \)[/tex], meaning [tex]\( y = \sqrt{x} + 7 \)[/tex].
2. Isolate [tex]\( x \)[/tex]: To find the inverse, solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. Start by isolating the square root term:
[tex]\[
y - 7 = \sqrt{x}
\][/tex]
3. Eliminate the square root: Square both sides of the equation to eliminate the square root:
[tex]\[
(y - 7)^2 = x
\][/tex]
4. Rewrite the equation: Express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
x = (y - 7)^2
\][/tex]
5. Express the inverse function: Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] in the equation to express the inverse function:
[tex]\[
f^{-1}(x) = (x - 7)^2
\][/tex]
6. Determine the domain of the inverse function: Since the original function [tex]\( f(x) = \sqrt{x} + 7 \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex] (because [tex]\( \sqrt{x} \)[/tex] is only real for non-negative [tex]\( x \)[/tex]), the inverse function [tex]\( f^{-1}(x) \)[/tex] will be defined for [tex]\( y \geq 7 \)[/tex] (since [tex]\( \sqrt{x} \)[/tex] starts at 0 and the minimum value of [tex]\( \sqrt{x} + 7 \)[/tex] is 7 when [tex]\( x = 0 \)[/tex]).
Therefore, the inverse function is [tex]\( f^{-1}(x) = (x - 7)^2 \)[/tex] with the domain [tex]\( x \geq 7 \)[/tex].
The correct answer is:
C. [tex]\( f^{-1}(x) = (x - 7)^2 \)[/tex], for [tex]\( x \geq 7 \)[/tex].
