Find g(f(x)) f(x)=3x-3 and g(x)=4x^2-3

$f (x) = 3 x - 3$ and $g (x) = 4 x^{2} - 3$

Set up the composite result function.

$g (f (x))$

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

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

Simplify each term.

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

$g (3 x - 3) = 4 ((3 x - 3) (3 x - 3)) - 3$

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

Apply the distributive property.

$g (3 x - 3) = 4 (3 x (3 x - 3) - 3 (3 x - 3)) - 3$

Apply the distributive property.

$g (3 x - 3) = 4 (3 x (3 x) + 3 x \cdot - 3 - 3 (3 x - 3)) - 3$

Apply the distributive property.

$g (3 x - 3) = 4 (3 x (3 x) + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3) - 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$g (3 x - 3) = 4 (3 \cdot (3 x \cdot x) + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3) - 3$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$g (3 x - 3) = 4 (3 \cdot (3 (x \cdot x)) + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3) - 3$

Multiply $x$ by $x$ .

$g (3 x - 3) = 4 (3 \cdot (3 x^{2}) + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3) - 3$

Multiply $3$ by $3$ .

$g (3 x - 3) = 4 (9 x^{2} + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3) - 3$

Multiply $- 3$ by $3$ .

$g (3 x - 3) = 4 (9 x^{2} - 9 x - 3 (3 x) - 3 \cdot - 3) - 3$

Multiply $3$ by $- 3$ .

$g (3 x - 3) = 4 (9 x^{2} - 9 x - 9 x - 3 \cdot - 3) - 3$

Multiply $- 3$ by $- 3$ .

$g (3 x - 3) = 4 (9 x^{2} - 9 x - 9 x + 9) - 3$

Subtract $9 x$ from $- 9 x$ .

$g (3 x - 3) = 4 (9 x^{2} - 18 x + 9) - 3$

Apply the distributive property.

$g (3 x - 3) = 4 (9 x^{2}) + 4 (- 18 x) + 4 \cdot 9 - 3$

Simplify.

Multiply $9$ by $4$ .

$g (3 x - 3) = 36 x^{2} + 4 (- 18 x) + 4 \cdot 9 - 3$

Multiply $- 18$ by $4$ .

$g (3 x - 3) = 36 x^{2} - 72 x + 4 \cdot 9 - 3$

Multiply $4$ by $9$ .

$g (3 x - 3) = 36 x^{2} - 72 x + 36 - 3$

Subtract $3$ from $36$ .

$g (3 x - 3) = 36 x^{2} - 72 x + 33$