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

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

Set up the composite result function.

$g (f (x))$

Evaluate $g (2 x - 4)$ by substituting in the value of $f$ into $g$ .

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

Simplify each term.

Rewrite ${(2 x - 4)}^{2}$ as $(2 x - 4) (2 x - 4)$ .

$g (2 x - 4) = 4 ((2 x - 4) (2 x - 4)) - 2$

Expand $(2 x - 4) (2 x - 4)$ using the FOIL Method.

Apply the distributive property.

$g (2 x - 4) = 4 (2 x (2 x - 4) - 4 (2 x - 4)) - 2$

Apply the distributive property.

$g (2 x - 4) = 4 (2 x (2 x) + 2 x \cdot - 4 - 4 (2 x - 4)) - 2$

Apply the distributive property.

$g (2 x - 4) = 4 (2 x (2 x) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4) - 2$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$g (2 x - 4) = 4 (2 \cdot (2 x \cdot x) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4) - 2$

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

Move $x$ .

$g (2 x - 4) = 4 (2 \cdot (2 (x \cdot x)) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4) - 2$

Multiply $x$ by $x$ .

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

Multiply $2$ by $2$ .

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

Multiply $- 4$ by $2$ .

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

Multiply $2$ by $- 4$ .

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

Multiply $- 4$ by $- 4$ .

$g (2 x - 4) = 4 (4 x^{2} - 8 x - 8 x + 16) - 2$

Subtract $8 x$ from $- 8 x$ .

$g (2 x - 4) = 4 (4 x^{2} - 16 x + 16) - 2$

Apply the distributive property.

$g (2 x - 4) = 4 (4 x^{2}) + 4 (- 16 x) + 4 \cdot 16 - 2$

Simplify.

Multiply $4$ by $4$ .

$g (2 x - 4) = 16 x^{2} + 4 (- 16 x) + 4 \cdot 16 - 2$

Multiply $- 16$ by $4$ .

$g (2 x - 4) = 16 x^{2} - 64 x + 4 \cdot 16 - 2$

Multiply $4$ by $16$ .

$g (2 x - 4) = 16 x^{2} - 64 x + 64 - 2$

Subtract $2$ from $64$ .

$g (2 x - 4) = 16 x^{2} - 64 x + 62$