We are given the function [tex]\( f(x) = x^2 \)[/tex] and need to determine which of the following options correctly represents the function [tex]\( g \)[/tex]:
A. [tex]\( g(x) = f(2x) \)[/tex]
B. [tex]\( g(x) = f(4x) \)[/tex]
C. [tex]\( g(x) = 2f(x) \)[/tex]
D. [tex]\( g(x) = f\left(\frac{1}{2}x\right) \)[/tex]
To solve this, let's examine each option individually:
### Option A: [tex]\( g(x) = f(2x) \)[/tex]
First, we replace [tex]\( 2x \)[/tex] in the function [tex]\( f \)[/tex]:
[tex]\[ g(x) = (2x)^2 = 4x^2 \][/tex]
### Option B: [tex]\( g(x) = f(4x) \)[/tex]
Next, we replace [tex]\( 4x \)[/tex] in the function [tex]\( f \)[/tex]:
[tex]\[ g(x) = (4x)^2 = 16x^2 \][/tex]
### Option C: [tex]\( g(x) = 2f(x) \)[/tex]
We plug [tex]\( x \)[/tex] into the function [tex]\( f \)[/tex] and multiply the result by 2:
[tex]\[ g(x) = 2 \cdot x^2 = 2x^2 \][/tex]
### Option D: [tex]\( g(x) = f\left(\frac{1}{2}x\right) \)[/tex]
Finally, we replace [tex]\( \frac{1}{2}x \)[/tex] in the function [tex]\( f \)[/tex]:
[tex]\[ g(x) = \left(\frac{1}{2}x\right)^2 = \frac{1}{4}x^2 \][/tex]
After examining all the options:
- Option A results in [tex]\( g(x) = 4x^2 \)[/tex].
- Option B results in [tex]\( g(x) = 16x^2 \)[/tex].
- Option C results in [tex]\( g(x) = 2x^2 \)[/tex].
- Option D results in [tex]\( g(x) = \frac{1}{4}x^2 \)[/tex].
We see that none of these exactly fits with the given information of [tex]\( g \)[/tex]. Thus, [tex]\( g \)[/tex] cannot be uniquely determined from the given choices.
