Find g(f(x)) f(x)=x+3 , g(x)=7x^2

$f (x) = x + 3$ , $g (x) = 7 x^{2}$

Set up the composite result function.

$g (f (x))$

Evaluate $g (x + 3)$ by substituting in the value of $f$ into $g$ .

$g (x + 3) = 7 {(x + 3)}^{2}$

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$g (x + 3) = 7 ((x + 3) (x + 3))$

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$g (x + 3) = 7 (x (x + 3) + 3 (x + 3))$

Apply the distributive property.

$g (x + 3) = 7 (x \cdot x + x \cdot 3 + 3 (x + 3))$

Apply the distributive property.

$g (x + 3) = 7 (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$g (x + 3) = 7 (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3)$

Move $3$ to the left of $x$ .

$g (x + 3) = 7 (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3)$

Multiply $3$ by $3$ .

$g (x + 3) = 7 (x^{2} + 3 x + 3 x + 9)$

Add $3 x$ and $3 x$ .

$g (x + 3) = 7 (x^{2} + 6 x + 9)$

Apply the distributive property.

$g (x + 3) = 7 x^{2} + 7 (6 x) + 7 \cdot 9$

Simplify.

Multiply $6$ by $7$ .

$g (x + 3) = 7 x^{2} + 42 x + 7 \cdot 9$

Multiply $7$ by $9$ .

$g (x + 3) = 7 x^{2} + 42 x + 63$