If $$ f(x)=3 x+2 $$ and $$ g(x)=x^{2}+1 $$, which expression is equivalent to $$ (f \circ g)(x) $$ ? $$ (3 x+2)\left(x^{2}+1\right) $$ $$ 3 x^{2}+1+2 $$ $$ (3 x+2)^{2}+1 $$ $$ 3\left(x^{2}+1\right)+2 $$

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UpStudy AI · Accepted Answer

To find the expression equivalent to $$ (f \circ g)(x) $$, we need to compute $$ f(g(x)) $$.

Given:
- $$ f(x) = 3x + 2 $$
- $$ g(x) = x^2 + 1 $$

Now, we will substitute $$ g(x) $$ into $$ f(x) $$:

1. Start with $$ g(x) $$:
   $$
   g(x) = x^2 + 1
   $$

2. Substitute $$ g(x) $$ into $$ f(x) $$:
   $$
   f(g(x)) = f(x^2 + 1) = 3(x^2 + 1) + 2
   $$

Now, let's simplify $$ 3(x^2 + 1) + 2 $$:

$$
3(x^2 + 1) + 2 = 3x^2 + 3 + 2 = 3x^2 + 5
$$

Now, let's compare this with the provided options:

1. $$ (3 x+2)(x^{2}+1) $$
2. $$ 3 x^{2}+1+2 $$
3. $$ (3 x+2)^{2}+1 $$
4. $$ 3(x^{2}+1)+2 $$

The expression $$ 3(x^{2}+1)+2 $$ simplifies to $$ 3x^2 + 5 $$, which matches our result.

Thus, the correct answer is:
$$
3(x^{2}+1)+2
$$
$$ 3(x^{2}+1)+2 $$

If and , which expression is equivalent to ?

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