Let `f:R to R` and `g:R to R` be given by `f(x)=3x^(2)+2 and g(x)=3x-1` for all `x to R`. Then,

Options

A. `fog(x)=27x^(2)-18x+5`
B. `fog(x)=27x^(2)+18x-5`
C. `gof(x)=9x^(2)-5`
D. `gof(x)=9x^(2)+15`

Answer

To solve the problem, we need to find the compositions \( f(g(x)) \) and \( g(f(x)) \) given the functions \( f(x) = 3x^2 + 2 \) and \( g(x) = 3x - 1 \).

### Step 1: Find \( f(g(x)) \)

1. Start with the function \( g(x) \):
\[ g(x) = 3x - 1 \]

2. Substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(3x - 1) \]

3. Now replace \( x \) in \( f(x) \) with \( 3x - 1 \):
\[ f(3x - 1) = 3(3x - 1)^2 + 2 \]

4. Expand \( (3x - 1)^2 \):
\[ (3x - 1)^2 = 9x^2 - 6x + 1 \]

5. Substitute back into the function:
\[ f(3x - 1) = 3(9x^2 - 6x + 1) + 2 \]

6. Distribute the 3:
\[ = 27x^2 - 18x + 3 + 2 \]

7. Combine like terms:
\[ = 27x^2 - 18x + 5 \]

Thus, we have:
\[ f(g(x)) = 27x^2 - 18x + 5 \]

### Step 2: Find \( g(f(x)) \)

1. Start with the function \( f(x) \):
\[ f(x) = 3x^2 + 2 \]

2. Substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(3x^2 + 2) \]

3. Now replace \( x \) in \( g(x) \) with \( 3x^2 + 2 \):
\[ g(3x^2 + 2) = 3(3x^2 + 2) - 1 \]

4. Distribute the 3:
\[ = 9x^2 + 6 - 1 \]

5. Combine like terms:
\[ = 9x^2 + 5 \]

Thus, we have:
\[ g(f(x)) = 9x^2 + 5 \]

### Final Results

We conclude with the results:
\[ f(g(x)) = 27x^2 - 18x + 5 \]
\[ g(f(x)) = 9x^2 + 5 \]