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

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

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

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

Step 3

Simplify each term.

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

$f (4 x - 2) = (4 x - 2) (4 x - 2) - 3$

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

Apply the distributive property.

$f (4 x - 2) = 4 x (4 x - 2) - 2 (4 x - 2) - 3$

Apply the distributive property.

$f (4 x - 2) = 4 x (4 x) + 4 x \cdot - 2 - 2 (4 x - 2) - 3$

Apply the distributive property.

$f (4 x - 2) = 4 x (4 x) + 4 x \cdot - 2 - 2 (4 x) - 2 \cdot - 2 - 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$f (4 x - 2) = 4 \cdot (4 x \cdot x) + 4 x \cdot - 2 - 2 (4 x) - 2 \cdot - 2 - 3$

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

Move $x$ .

$f (4 x - 2) = 4 \cdot (4 (x \cdot x)) + 4 x \cdot - 2 - 2 (4 x) - 2 \cdot - 2 - 3$

Multiply $x$ by $x$ .

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

Multiply $4$ by $4$ .

$f (4 x - 2) = 16 x^{2} + 4 x \cdot - 2 - 2 (4 x) - 2 \cdot - 2 - 3$

Multiply $- 2$ by $4$ .

$f (4 x - 2) = 16 x^{2} - 8 x - 2 (4 x) - 2 \cdot - 2 - 3$

Multiply $4$ by $- 2$ .

$f (4 x - 2) = 16 x^{2} - 8 x - 8 x - 2 \cdot - 2 - 3$

Multiply $- 2$ by $- 2$ .

$f (4 x - 2) = 16 x^{2} - 8 x - 8 x + 4 - 3$

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

$f (4 x - 2) = 16 x^{2} - 16 x + 4 - 3$

Step 4

Subtract $3$ from $4$ .

$f (4 x - 2) = 16 x^{2} - 16 x + 1$