Let `f:R to ` 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)`

Answer

To find \( g \circ f \), we need to evaluate \( g(f(x)) \). Given the functions:

1. \( f(x) = 3x^2 - 5 \)
2. \( g(x) = \frac{x}{x^2 + 1} \)

We will substitute \( f(x) \) into \( g(x) \).

### Step 1: Substitute \( f(x) \) into \( g(x) \)

We start by substituting \( f(x) \) into \( g(x) \):

\[ g(f(x)) = g(3x^2 - 5) \]

### Step 2: Write the expression for \( g(f(x)) \)

Now, we replace \( x \) in \( g(x) \) with \( f(x) \):

\[ g(3x^2 - 5) = \frac{3x^2 - 5}{(3x^2 - 5)^2 + 1} \]

### Step 3: Simplify the denominator

Next, we need to simplify the denominator \( (3x^2 - 5)^2 + 1 \):

\[ (3x^2 - 5)^2 = (3x^2)^2 - 2 \cdot 3x^2 \cdot 5 + 5^2 = 9x^4 - 30x^2 + 25 \]

So, adding 1:

\[ (3x^2 - 5)^2 + 1 = 9x^4 - 30x^2 + 25 + 1 = 9x^4 - 30x^2 + 26 \]

### Step 4: Write the final expression for \( g(f(x)) \)

Now, we can write the complete expression for \( g(f(x)) \):

\[ g(f(x)) = \frac{3x^2 - 5}{9x^4 - 30x^2 + 26} \]

### Final Result

Thus, the composition \( g \circ f \) is:

\[ g \circ f = \frac{3x^2 - 5}{9x^4 - 30x^2 + 26} \]

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