To find the inverse of the function [tex]\( f(x) = (2x - 4)^2 \)[/tex] for [tex]\( x \geq 2 \)[/tex], we follow these steps:
1. Start with the given function:
[tex]\[ f(x) = (2x - 4)^2 \][/tex]
2. Let [tex]\( y = f(x) \)[/tex]:
[tex]\[ y = (2x - 4)^2 \][/tex]
3. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
To do this, we take the square root of both sides of the equation.
[tex]\[ \sqrt{y} = 2x - 4 \][/tex]
4. Isolate [tex]\( x \)[/tex]:
Add 4 to both sides.
[tex]\[ \sqrt{y} + 4 = 2x \][/tex]
Now, divide by 2.
[tex]\[ x = \frac{\sqrt{y} + 4}{2} \][/tex]
Since [tex]\( y \)[/tex] is the output of the original function and [tex]\( x \)[/tex] is the input, we can express the inverse function [tex]\( g \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ g(x) = \frac{\sqrt{x} + 4}{2} \][/tex]
Now, we match this function to one of the given options:
[tex]\[ g(x) = \frac{1}{2} \sqrt{x} + 2, \quad x \geq 0 \][/tex]
Thus, the inverse of [tex]\( f(x) = (2x - 4)^2 \)[/tex] for [tex]\( x \geq 2 \)[/tex] is:
[tex]\[ g(x) = \frac{1}{2} \sqrt{x} + 2, \quad x \geq 0 \][/tex]
The correct answer is:
[tex]\[ \boxed{\frac{1}{2} \sqrt{x} + 2, \quad x \geq 0} \][/tex]
