Answer :
To determine if [tex]\( g(x) = \frac{1}{3}x \)[/tex] is indeed the inverse of [tex]\( f(x) = 3x \)[/tex], we need to verify the compositions [tex]\( g(f(x)) \)[/tex] and [tex]\( f(g(x)) \)[/tex]. Specifically, each composition should yield the identity function [tex]\( x \)[/tex].
### Step-by-Step Verification
#### 1. Verify [tex]\( f(g(x)) \)[/tex]
First, let's compute [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = f\left(\frac{1}{3}x\right) = 3 \left(\frac{1}{3}x\right) = \left(\frac{3}{3}\right)x = x \][/tex]
The expression [tex]\( \frac{1}{3}(3x) \)[/tex] verifies that [tex]\( f(g(x)) = x \)[/tex]. This matches the identity function, verifying [tex]\( g(x) \)[/tex] as the inverse of [tex]\( f(x) \)[/tex] in this case.
#### 2. Verify [tex]\( g(f(x)) \)[/tex]
Next, let's compute [tex]\( g(f(x)) \)[/tex]:
[tex]\[ g(f(x)) = g(3x) = \frac{1}{3}(3x) = \left(\frac{3}{3}\right)x = x \][/tex]
The expression [tex]\( g(3x) = \frac{1}{3}(3x) \)[/tex] also gives us [tex]\( x \)[/tex], satisfying the identity function, which verifies [tex]\( f(x) \)[/tex] as the inverse of [tex]\( g(x) \)[/tex].
### Summary of the Options Provided
Now we can evaluate the given expressions to see which ones confirm [tex]\( g(x) = f^{-1}(x) \)[/tex].
1. [tex]\( 3x \left( \frac{x}{3} \right) \)[/tex]:
[tex]\[ 3x \left( \frac{x}{3} \right) = 3x \cdot \frac{x}{3} = x^2 \][/tex]
This results in [tex]\( x^2 \)[/tex], not [tex]\( x \)[/tex]. Therefore, it is incorrect.
2. [tex]\( \frac{1}{3}x \cdot 3x \)[/tex]:
[tex]\[ \frac{1}{3}x \cdot 3x = \left(\frac{1}{3} \cdot 3\right)x^2 = x^2 \][/tex]
This also results in [tex]\( x^2 \)[/tex], not [tex]\( x \)[/tex]. Therefore, it is incorrect.
3. [tex]\( \frac{1}{3}(3x) \)[/tex]:
[tex]\[ \frac{1}{3}(3x) = \left(\frac{3}{3}\right)x = x \][/tex]
This results in [tex]\( x \)[/tex], confirming it as a valid check.
4. [tex]\( \frac{1}{3}\left(\frac{1}{3}x\right) \)[/tex]:
[tex]\[ \frac{1}{3}\left(\frac{1}{3}x\right) = \frac{1}{9}x \][/tex]
This results in [tex]\( \frac{1}{9} x \)[/tex], not [tex]\( x \)[/tex]. Therefore, it is incorrect.
### Conclusion
The correct expressions that verify [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex] are:
- [tex]\( \frac{1}{3}(3x) \)[/tex]
- [tex]\( \left(\frac{1}{3} x \right)(3 x) \)[/tex]
Thus, the correct options are the second and third expressions:
[tex]\[ \boxed{\left(\frac{1}{3}x\right)(3x), \frac{1}{3}(3x)} \][/tex]
### Step-by-Step Verification
#### 1. Verify [tex]\( f(g(x)) \)[/tex]
First, let's compute [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = f\left(\frac{1}{3}x\right) = 3 \left(\frac{1}{3}x\right) = \left(\frac{3}{3}\right)x = x \][/tex]
The expression [tex]\( \frac{1}{3}(3x) \)[/tex] verifies that [tex]\( f(g(x)) = x \)[/tex]. This matches the identity function, verifying [tex]\( g(x) \)[/tex] as the inverse of [tex]\( f(x) \)[/tex] in this case.
#### 2. Verify [tex]\( g(f(x)) \)[/tex]
Next, let's compute [tex]\( g(f(x)) \)[/tex]:
[tex]\[ g(f(x)) = g(3x) = \frac{1}{3}(3x) = \left(\frac{3}{3}\right)x = x \][/tex]
The expression [tex]\( g(3x) = \frac{1}{3}(3x) \)[/tex] also gives us [tex]\( x \)[/tex], satisfying the identity function, which verifies [tex]\( f(x) \)[/tex] as the inverse of [tex]\( g(x) \)[/tex].
### Summary of the Options Provided
Now we can evaluate the given expressions to see which ones confirm [tex]\( g(x) = f^{-1}(x) \)[/tex].
1. [tex]\( 3x \left( \frac{x}{3} \right) \)[/tex]:
[tex]\[ 3x \left( \frac{x}{3} \right) = 3x \cdot \frac{x}{3} = x^2 \][/tex]
This results in [tex]\( x^2 \)[/tex], not [tex]\( x \)[/tex]. Therefore, it is incorrect.
2. [tex]\( \frac{1}{3}x \cdot 3x \)[/tex]:
[tex]\[ \frac{1}{3}x \cdot 3x = \left(\frac{1}{3} \cdot 3\right)x^2 = x^2 \][/tex]
This also results in [tex]\( x^2 \)[/tex], not [tex]\( x \)[/tex]. Therefore, it is incorrect.
3. [tex]\( \frac{1}{3}(3x) \)[/tex]:
[tex]\[ \frac{1}{3}(3x) = \left(\frac{3}{3}\right)x = x \][/tex]
This results in [tex]\( x \)[/tex], confirming it as a valid check.
4. [tex]\( \frac{1}{3}\left(\frac{1}{3}x\right) \)[/tex]:
[tex]\[ \frac{1}{3}\left(\frac{1}{3}x\right) = \frac{1}{9}x \][/tex]
This results in [tex]\( \frac{1}{9} x \)[/tex], not [tex]\( x \)[/tex]. Therefore, it is incorrect.
### Conclusion
The correct expressions that verify [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex] are:
- [tex]\( \frac{1}{3}(3x) \)[/tex]
- [tex]\( \left(\frac{1}{3} x \right)(3 x) \)[/tex]
Thus, the correct options are the second and third expressions:
[tex]\[ \boxed{\left(\frac{1}{3}x\right)(3x), \frac{1}{3}(3x)} \][/tex]
