Find f(g(x)) f(x)=2x^2-4x-3 , g(x)=2x^2-16

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (2 x^{2} - 16)$ by substituting in the value of $g$ into $f$ .

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

Step 3

Simplify each term.

Rewrite ${(2 x^{2} - 16)}^{2}$ as $(2 x^{2} - 16) (2 x^{2} - 16)$ .

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

Expand $(2 x^{2} - 16) (2 x^{2} - 16)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $2$ and $2$ .

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

Multiply $2$ by $2$ .

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

Multiply $- 16$ by $2$ .

$f (2 x^{2} - 16) = 2 (4 x^{4} - 32 x^{2} - 16 (2 x^{2}) - 16 \cdot - 16) - 4 (2 x^{2} - 16) - 3$

Multiply $2$ by $- 16$ .

$f (2 x^{2} - 16) = 2 (4 x^{4} - 32 x^{2} - 32 x^{2} - 16 \cdot - 16) - 4 (2 x^{2} - 16) - 3$

Multiply $- 16$ by $- 16$ .

$f (2 x^{2} - 16) = 2 (4 x^{4} - 32 x^{2} - 32 x^{2} + 256) - 4 (2 x^{2} - 16) - 3$

Subtract $32 x^{2}$ from $- 32 x^{2}$ .

$f (2 x^{2} - 16) = 2 (4 x^{4} - 64 x^{2} + 256) - 4 (2 x^{2} - 16) - 3$

Apply the distributive property.

$f (2 x^{2} - 16) = 2 (4 x^{4}) + 2 (- 64 x^{2}) + 2 \cdot 256 - 4 (2 x^{2} - 16) - 3$

Simplify.

Multiply $4$ by $2$ .

$f (2 x^{2} - 16) = 8 x^{4} + 2 (- 64 x^{2}) + 2 \cdot 256 - 4 (2 x^{2} - 16) - 3$

Multiply $- 64$ by $2$ .

$f (2 x^{2} - 16) = 8 x^{4} - 128 x^{2} + 2 \cdot 256 - 4 (2 x^{2} - 16) - 3$

Multiply $2$ by $256$ .

$f (2 x^{2} - 16) = 8 x^{4} - 128 x^{2} + 512 - 4 (2 x^{2} - 16) - 3$

Apply the distributive property.

$f (2 x^{2} - 16) = 8 x^{4} - 128 x^{2} + 512 - 4 (2 x^{2}) - 4 \cdot - 16 - 3$

Multiply $2$ by $- 4$ .

$f (2 x^{2} - 16) = 8 x^{4} - 128 x^{2} + 512 - 8 x^{2} - 4 \cdot - 16 - 3$

Multiply $- 4$ by $- 16$ .

$f (2 x^{2} - 16) = 8 x^{4} - 128 x^{2} + 512 - 8 x^{2} + 64 - 3$

Step 4

Simplify by adding terms.

Subtract $8 x^{2}$ from $- 128 x^{2}$ .

$f (2 x^{2} - 16) = 8 x^{4} - 136 x^{2} + 512 + 64 - 3$

Simplify by adding and subtracting.

Add $512$ and $64$ .

$f (2 x^{2} - 16) = 8 x^{4} - 136 x^{2} + 576 - 3$

Subtract $3$ from $576$ .

$f (2 x^{2} - 16) = 8 x^{4} - 136 x^{2} + 573$

Find f(g(x)) f(x)=2x^2-4x-3 , g(x)=2x^2-16

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