To verify that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex], we need to confirm two conditions:
1. [tex]\( f(g(x)) = x \)[/tex]
2. [tex]\( g(f(x)) = x \)[/tex]
Let's evaluate each composition step-by-step to identify which expressions match these conditions.
Step 1: Finding [tex]\( f(g(x)) \)[/tex]
Given [tex]\( f(x) = 3x \)[/tex] and [tex]\( g(x) = \frac{1}{3}x \)[/tex], we first find [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = f\left(\frac{1}{3} x\right) \][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f\left(\frac{1}{3} x\right) = 3 \left(\frac{1}{3} x\right) \][/tex]
Simplify the expression inside [tex]\( f \)[/tex]:
[tex]\[ 3 \left(\frac{1}{3} x\right) = \left(3 \cdot \frac{1}{3}\right) x = 1 \cdot x = x \][/tex]
Thus, [tex]\( f(g(x)) = x \)[/tex] verifies one of the conditions. The matching expression from the choices provided is:
[tex]\[ 3 \left(\frac{x}{3}\right) \][/tex]
Step 2: Finding [tex]\( g(f(x)) \)[/tex]
Next, we find [tex]\( g(f(x)) \)[/tex]:
[tex]\[ g(f(x)) = g(3x) \][/tex]
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(3x) = \frac{1}{3} (3x) \][/tex]
Simplify the expression inside [tex]\( g \)[/tex]:
[tex]\[ \frac{1}{3} (3x) = \left(\frac{1}{3} \cdot 3\right) x = 1 \cdot x = x \][/tex]
Thus, [tex]\( g(f(x)) = x \)[/tex] verifies the second condition. The matching expression from the choices provided is:
[tex]\[ \frac{1}{3}(3 x) \][/tex]
Therefore, the following two expressions can be used to verify that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex]:
1. [tex]\( 3 x\left(\frac{x}{3}\right) \)[/tex]
2. [tex]\( \frac{1}{3}(3 x) \)[/tex]
Thus, the correct expressions from the choices are:
[tex]\[3 x\left(\frac{x}{3}\right) \quad \text{and} \quad \frac{1}{3}(3 x)\][/tex]
