Evaluate the Piecewise Function f(x)=2x , g(x)=3x^2+6 , g(f(2))

$f (x) = 2 x$ , $g (x) = 3 x^{2} + 6$ , $g (f (2))$

Set up the composite result function.

$g (f (x))$

Evaluate $g (2 x)$ by substituting in the value of $f$ into $g$ .

$g (2 x) = 3 {(2 x)}^{2} + 6$

Simplify each term.

Apply the product rule to $2 x$ .

$g (2 x) = 3 (2^{2} x^{2}) + 6$

Raise $2$ to the power of $2$ .

$g (2 x) = 3 (4 x^{2}) + 6$

Multiply $4$ by $3$ .

$g (2 x) = 12 x^{2} + 6$

Evaluate the result function by replacing $x$ with $2$ in the function.

$12 {(2)}^{2} + 6$

Simplify the evaluated function.

Simplify each term.

Raise $2$ to the power of $2$ .

$12 \cdot 4 + 6$

Multiply $12$ by $4$ .

$48 + 6$

Add $48$ and $6$ .

$54$