To verify that [tex]\( g(x) = \frac{1}{3} x \)[/tex] is the inverse of [tex]\( f(x) = 3x \)[/tex], we need to check two conditions:
1. [tex]\( f(g(x)) = x \)[/tex]
2. [tex]\( g(f(x)) = x \)[/tex]
Let's analyze each given option one by one:
### Option 1: [tex]\( 3x \left(\frac{x}{3}\right) \)[/tex]
Here, the expression is:
[tex]\[ 3x \left(\frac{x}{3}\right) \][/tex]
- Substitute [tex]\( g(x) = \frac{1}{3} x \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = 3 \left(\frac{x}{3}\right) \][/tex]
- Simplify:
[tex]\[ f(g(x)) = x \][/tex]
This correctly checks the first condition, [tex]\( f(g(x)) = x \)[/tex].
### Option 2: [tex]\( \left(\frac{1}{3} x\right)(3x) \)[/tex]
Here, the expression is:
[tex]\[ \left(\frac{1}{3} x\right)(3x) \][/tex]
- Substitute [tex]\( f(x) = 3x \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = \frac{1}{3} (3x) \][/tex]
- Simplify:
[tex]\[ g(f(x)) = x \][/tex]
This correctly checks the second condition, [tex]\( g(f(x)) = x \)[/tex].
### Option 3: [tex]\( \frac{1}{3}(3x) \)[/tex]
Here, the expression is:
[tex]\[ \frac{1}{3}(3x) \][/tex]
- This corresponds to [tex]\( g(f(x)) \)[/tex] as:
[tex]\[ g(f(x)) = \frac{1}{3} (3x) \][/tex]
- Simplify:
[tex]\[ g(f(x)) = x \][/tex]
This correctly checks the second condition, [tex]\( g(f(x)) = x \)[/tex].
### Option 4: [tex]\( \frac{1}{3}\left(\frac{1}{3}x\right) \)[/tex]
Here, the expression is:
[tex]\[ \frac{1}{3}\left(\frac{1}{3} x\right) \][/tex]
- This does not align with the forms of [tex]\( f(g(x)) \)[/tex] or [tex]\( g(f(x)) \)[/tex] we need to check, thus it is not relevant to verifying the inverse relationship between [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
Therefore, the correct expressions to verify that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex] are:
- [tex]\( 3x\left(\frac{x}{3}\right) \)[/tex]
- [tex]\( \left(\frac{1}{3}x\right)(3x) \)[/tex]
Hence, the correct expressions for verifying the inverse are:
[tex]\[ 3x\left(\frac{x}{3}\right) \][/tex]
and
[tex]\[ \left(\frac{1}{3}x\right)(3x) \][/tex]
Thus, the correct choice in this context is [tex]\([1, 2]\)[/tex].
