If `f:R to R` be defined by `f(x)=3x^(2)-5` and `g: R to R ` by `g(x)= (x)/(x^(2)+1).` Then, gof is

Options

A. `(3x^(2)-5)/(9x^(4)-30x^(2)+26)`
B. `(3x^(2)-5)/(9x^(4)-6x^(2)+26)`
C. `(3x^(2))/(x^(4)+2x^(2)-4)`
D. `(3x^(2))/(9x^(4)+30x^(2)-2)`

Explanation

To find \( g \circ f \), we need to compute \( g(f(x)) \).

### Step-by-Step Solution:

1. **Identify the functions**:
- The function \( f \) is given by:
\[ f(x) = 3x^2 - 5 \]
- The function \( g \) is defined as:
\[ g(x) = \frac{x}{x^2 + 1} \]

2. **Substitute \( f(x) \) into \( g(x) \)**:
- We need to find \( g(f(x)) \), which means substituting \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(3x^2 - 5) \]

3. **Replace \( x \) in \( g(x) \) with \( f(x) \)**:
- Now substitute \( 3x^2 - 5 \) into the function \( g \):
\[ g(3x^2 - 5) = \frac{3x^2 - 5}{(3x^2 - 5)^2 + 1} \]

4. **Calculate \( (3x^2 - 5)^2 + 1 \)**:
- First, calculate \( (3x^2 - 5)^2 \):
\[ (3x^2 - 5)^2 = 9x^4 - 30x^2 + 25 \]
- Now add 1:
\[ (3x^2 - 5)^2 + 1 = 9x^4 - 30x^2 + 25 + 1 = 9x^4 - 30x^2 + 26 \]

5. **Combine the results**:
- Now we can write \( g(f(x)) \):
\[ g(f(x)) = \frac{3x^2 - 5}{9x^4 - 30x^2 + 26} \]

6. **Final Result**:
- Therefore, the composition \( g \circ f \) is:
\[ g \circ f = \frac{3x^2 - 5}{9x^4 - 30x^2 + 26} \]