Find g(f(x)) f(x)=2x-2 and g(x)=6x^2-3

$f (x) = 2 x - 2$ and $g (x) = 6 x^{2} - 3$

Set up the composite result function.

$g (f (x))$

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

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

Simplify each term.

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

$g (2 x - 2) = 6 ((2 x - 2) (2 x - 2)) - 3$

Expand $(2 x - 2) (2 x - 2)$ using the FOIL Method.

Apply the distributive property.

$g (2 x - 2) = 6 (2 x (2 x - 2) - 2 (2 x - 2)) - 3$

Apply the distributive property.

$g (2 x - 2) = 6 (2 x (2 x) + 2 x \cdot - 2 - 2 (2 x - 2)) - 3$

Apply the distributive property.

$g (2 x - 2) = 6 (2 x (2 x) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2) - 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$g (2 x - 2) = 6 (2 \cdot (2 x \cdot x) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2) - 3$

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

Move $x$ .

$g (2 x - 2) = 6 (2 \cdot (2 (x \cdot x)) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2) - 3$

Multiply $x$ by $x$ .

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

Multiply $2$ by $2$ .

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

Multiply $- 2$ by $2$ .

$g (2 x - 2) = 6 (4 x^{2} - 4 x - 2 (2 x) - 2 \cdot - 2) - 3$

Multiply $2$ by $- 2$ .

$g (2 x - 2) = 6 (4 x^{2} - 4 x - 4 x - 2 \cdot - 2) - 3$

Multiply $- 2$ by $- 2$ .

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

Subtract $4 x$ from $- 4 x$ .

$g (2 x - 2) = 6 (4 x^{2} - 8 x + 4) - 3$

Apply the distributive property.

$g (2 x - 2) = 6 (4 x^{2}) + 6 (- 8 x) + 6 \cdot 4 - 3$

Simplify.

Multiply $4$ by $6$ .

$g (2 x - 2) = 24 x^{2} + 6 (- 8 x) + 6 \cdot 4 - 3$

Multiply $- 8$ by $6$ .

$g (2 x - 2) = 24 x^{2} - 48 x + 6 \cdot 4 - 3$

Multiply $6$ by $4$ .

$g (2 x - 2) = 24 x^{2} - 48 x + 24 - 3$

Subtract $3$ from $24$ .

$g (2 x - 2) = 24 x^{2} - 48 x + 21$