To determine which equation represents the function [tex]\( g(x) \)[/tex] based on the definition of [tex]\( f(x) = x^2 \)[/tex], we will analyze the given options step by step.
Let's test a specific value of [tex]\( x \)[/tex], say [tex]\( x = 3 \)[/tex], and compare the results from each function.
First, compute [tex]\( f(x) \)[/tex] for [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 3^2 = 9 \][/tex]
Now, we evaluate each candidate function [tex]\( g(x) \)[/tex] at [tex]\( x = 3 \)[/tex]:
### Option A: [tex]\( g(x) = \frac{1}{3} f(x) \)[/tex]
[tex]\[ g(3) = \frac{1}{3} f(3) = \frac{1}{3} \times 9 = 3.0 \][/tex]
### Option B: [tex]\( g(x) = 3 f(x) \)[/tex]
[tex]\[ g(3) = 3 f(3) = 3 \times 9 = 27 \][/tex]
### Option C: [tex]\( g(x) = f\left(\frac{1}{3} x\right) \)[/tex]
[tex]\[ g(3) = f\left(\frac{1}{3} \times 3\right) = f(1) = 1^2 = 1.0 \][/tex]
### Option D: [tex]\( g(x) = f(3x) \)[/tex]
[tex]\[ g(3) = f(3 \times 3) = f(9) = 9^2 = 81 \][/tex]
Let's compare the results from our evaluations:
- From Option A, [tex]\( g(3) = 3.0 \)[/tex]
- From Option B, [tex]\( g(3) = 27 \)[/tex]
- From Option C, [tex]\( g(3) = 1.0 \)[/tex]
- From Option D, [tex]\( g(3) = 81 \)[/tex]
The results obtained are:
[tex]\[ (f(3), g_A(3), g_B(3), g_C(3), g_D(3)) = (9, 3.0, 27, 1.0, 81) \][/tex]
Based on these results, the transformations for [tex]\( f(x) \)[/tex] and how they affect the value at [tex]\( x = 3 \)[/tex], the function [tex]\( g(x) \)[/tex] that matches the transformations provided corresponds to:
- Option A: [tex]\( g(x) = \frac{1}{3} f(x) \Rightarrow g(3) = 3.0 \)[/tex]
- Option B: [tex]\( g(x) = 3 f(x) \Rightarrow g(3) = 27 \)[/tex]
- Option C: [tex]\( g(x) = f\left(\frac{1}{3} x\right) \Rightarrow g(3) = 1.0 \)[/tex]
- Option D: [tex]\( g(x) = f(3x) \Rightarrow g(3) = 81 \)[/tex]
Thus, each option corresponds to a different transformation:
- For [tex]\( g(x) = \frac{1}{3} f(x) \)[/tex], [tex]\( g \)[/tex] scales [tex]\( f(x) \)[/tex] by [tex]\(\frac{1}{3}\)[/tex].
- For [tex]\( g(x) = 3 f(x) \)[/tex], [tex]\( g \)[/tex] scales [tex]\( f(x) \)[/tex] by [tex]\(3\)[/tex].
- For [tex]\( g(x) = f\left(\frac{1}{3} x\right) \)[/tex], [tex]\( g \)[/tex] evaluates [tex]\( f \)[/tex] at [tex]\(\frac{1}{3}x\)[/tex].
- For [tex]\( g(x) = f(3x) \)[/tex], [tex]\( g \)[/tex] evaluates [tex]\( f \)[/tex] at [tex]\(3x\)[/tex].
These steps allow us to see how each function [tex]\( g(x) \)[/tex] is derived from [tex]\( f(x) \)[/tex] and which option corresponds to which function.
