To find the inverse of the function [tex]\( f(x) = (2x - 4)^2 \)[/tex] for [tex]\( x \geq 2 \)[/tex], follow these steps:
1. Set the function equal to [tex]\( y \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[
y = (2x - 4)^2
\][/tex]
2. Take the square root of both sides to solve for [tex]\( 2x - 4 \)[/tex]:
[tex]\[
\sqrt{y} = |2x - 4|
\][/tex]
Since [tex]\( x \geq 2 \)[/tex], [tex]\( 2x - 4 \geq 0 \)[/tex]. So we can remove the absolute value:
[tex]\[
\sqrt{y} = 2x - 4
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
2x - 4 = \sqrt{y}
\][/tex]
[tex]\[
2x = \sqrt{y} + 4
\][/tex]
[tex]\[
x = \frac{\sqrt{y} + 4}{2}
\][/tex]
4. Rewrite [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. Since [tex]\( g(y) \)[/tex] is the inverse function:
[tex]\[
g(x) = \frac{\sqrt{x} + 4}{2}
\][/tex]
Therefore, the correct inverse function [tex]\( g(x) \)[/tex] is:
[tex]\[
g(x) = \frac{1}{2}\sqrt{x} + 2, \quad x \geq 0
\][/tex]
Thus, the correct option is:
[tex]\[
g(x)=\frac{1}{2} \sqrt{x}+2, x \geq 0
\][/tex]
