If `f:RtoR` is defined by `f(x)=3x^(2)-5` and `g:RtoR` is defined by `g(x)=(x)/(x^(2)+1)` then (gof)(x) 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)`

Answer

To find \( (g \circ f)(x) \), we need to first understand the functions \( f(x) \) and \( g(x) \) given in the problem.

1. **Define the functions**:
- \( f(x) = 3x^2 - 5 \)
- \( g(x) = \frac{x}{x^2 + 1} \)

2. **Find \( g(f(x)) \)**:
- We substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(3x^2 - 5) \]
- Now, replace \( x \) in \( g(x) \) with \( 3x^2 - 5 \):
\[ g(3x^2 - 5) = \frac{3x^2 - 5}{(3x^2 - 5)^2 + 1} \]

3. **Simplify the denominator**:
- First, calculate \( (3x^2 - 5)^2 \):
\[ (3x^2 - 5)^2 = 9x^4 - 30x^2 + 25 \]
- Now add 1 to this result:
\[ (3x^2 - 5)^2 + 1 = 9x^4 - 30x^2 + 25 + 1 = 9x^4 - 30x^2 + 26 \]

4. **Combine the results**:
- Now substitute back into the expression for \( g(f(x)) \):
\[ g(f(x)) = \frac{3x^2 - 5}{9x^4 - 30x^2 + 26} \]

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