Answered

What is the inverse of [tex]f(x) = (2x - 4)^2[/tex] for [tex]x \geq 2[/tex] where function [tex]g[/tex] is the inverse of function [tex]f[/tex]?

A. [tex]g(x) = \frac{1}{2} \sqrt{x} + 2, x \geq 0[/tex]
B. [tex]g(x) = \frac{1}{2} \sqrt{x} - 2, x \geq 0[/tex]
C. [tex]g(x) = \sqrt{2x + 4}, x \geq 2[/tex]
D. [tex]g(x) = \sqrt{2x - 4}, x \geq 2[/tex]



Answer :

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]