Find g(f(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.

$g (f (x))$

Step 2

Evaluate $g (2 x^{2} - 4 x - 3)$ by substituting in the value of $f$ into $g$ .

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

Step 3

Simplify each term.

Rewrite ${(2 x^{2} - 4 x - 3)}^{2}$ as $(2 x^{2} - 4 x - 3) (2 x^{2} - 4 x - 3)$ .

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

Expand $(2 x^{2} - 4 x - 3) (2 x^{2} - 4 x - 3)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Multiply $2$ by $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

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

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

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} + 2 \cdot (- 4 x^{1 + 2}) + 2 x^{2} \cdot - 3 - 4 x (2 x^{2}) - 4 x (- 4 x) - 4 x \cdot - 3 - 3 (2 x^{2}) - 3 (- 4 x) - 3 \cdot - 3) - 16$

Add $1$ and $2$ .

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

Multiply $2$ by $- 4$ .

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

Multiply $- 3$ by $2$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 4 x (2 x^{2}) - 4 x (- 4 x) - 4 x \cdot - 3 - 3 (2 x^{2}) - 3 (- 4 x) - 3 \cdot - 3) - 16$

Rewrite using the commutative property of multiplication.

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 4 \cdot (2 x \cdot x^{2}) - 4 x (- 4 x) - 4 x \cdot - 3 - 3 (2 x^{2}) - 3 (- 4 x) - 3 \cdot - 3) - 16$

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

Move $x^{2}$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 4 \cdot (2 (x^{2} x)) - 4 x (- 4 x) - 4 x \cdot - 3 - 3 (2 x^{2}) - 3 (- 4 x) - 3 \cdot - 3) - 16$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 4 \cdot (2 (x^{2} x)) - 4 x (- 4 x) - 4 x \cdot - 3 - 3 (2 x^{2}) - 3 (- 4 x) - 3 \cdot - 3) - 16$

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

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 4 \cdot (2 x^{2 + 1}) - 4 x (- 4 x) - 4 x \cdot - 3 - 3 (2 x^{2}) - 3 (- 4 x) - 3 \cdot - 3) - 16$

Add $2$ and $1$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 4 \cdot (2 x^{3}) - 4 x (- 4 x) - 4 x \cdot - 3 - 3 (2 x^{2}) - 3 (- 4 x) - 3 \cdot - 3) - 16$

Multiply $- 4$ by $2$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 8 x^{3} - 4 x (- 4 x) - 4 x \cdot - 3 - 3 (2 x^{2}) - 3 (- 4 x) - 3 \cdot - 3) - 16$

Rewrite using the commutative property of multiplication.

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 8 x^{3} - 4 \cdot (- 4 x \cdot x) - 4 x \cdot - 3 - 3 (2 x^{2}) - 3 (- 4 x) - 3 \cdot - 3) - 16$

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

Move $x$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 8 x^{3} - 4 \cdot (- 4 (x \cdot x)) - 4 x \cdot - 3 - 3 (2 x^{2}) - 3 (- 4 x) - 3 \cdot - 3) - 16$

Multiply $x$ by $x$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 8 x^{3} - 4 \cdot (- 4 x^{2}) - 4 x \cdot - 3 - 3 (2 x^{2}) - 3 (- 4 x) - 3 \cdot - 3) - 16$

Multiply $- 4$ by $- 4$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 8 x^{3} + 16 x^{2} - 4 x \cdot - 3 - 3 (2 x^{2}) - 3 (- 4 x) - 3 \cdot - 3) - 16$

Multiply $- 3$ by $- 4$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 8 x^{3} + 16 x^{2} + 12 x - 3 (2 x^{2}) - 3 (- 4 x) - 3 \cdot - 3) - 16$

Multiply $2$ by $- 3$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 8 x^{3} + 16 x^{2} + 12 x - 6 x^{2} - 3 (- 4 x) - 3 \cdot - 3) - 16$

Multiply $- 4$ by $- 3$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 8 x^{3} + 16 x^{2} + 12 x - 6 x^{2} + 12 x - 3 \cdot - 3) - 16$

Multiply $- 3$ by $- 3$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 8 x^{3} - 6 x^{2} - 8 x^{3} + 16 x^{2} + 12 x - 6 x^{2} + 12 x + 9) - 16$

Subtract $8 x^{3}$ from $- 8 x^{3}$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 16 x^{3} - 6 x^{2} + 16 x^{2} + 12 x - 6 x^{2} + 12 x + 9) - 16$

Add $- 6 x^{2}$ and $16 x^{2}$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 16 x^{3} + 10 x^{2} + 12 x - 6 x^{2} + 12 x + 9) - 16$

Subtract $6 x^{2}$ from $10 x^{2}$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 16 x^{3} + 4 x^{2} + 12 x + 12 x + 9) - 16$

Add $12 x$ and $12 x$ .

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4} - 16 x^{3} + 4 x^{2} + 24 x + 9) - 16$

Apply the distributive property.

$g (2 x^{2} - 4 x - 3) = 2 (4 x^{4}) + 2 (- 16 x^{3}) + 2 (4 x^{2}) + 2 (24 x) + 2 \cdot 9 - 16$

Simplify.

Multiply $4$ by $2$ .

$g (2 x^{2} - 4 x - 3) = 8 x^{4} + 2 (- 16 x^{3}) + 2 (4 x^{2}) + 2 (24 x) + 2 \cdot 9 - 16$

Multiply $- 16$ by $2$ .

$g (2 x^{2} - 4 x - 3) = 8 x^{4} - 32 x^{3} + 2 (4 x^{2}) + 2 (24 x) + 2 \cdot 9 - 16$

Multiply $4$ by $2$ .

$g (2 x^{2} - 4 x - 3) = 8 x^{4} - 32 x^{3} + 8 x^{2} + 2 (24 x) + 2 \cdot 9 - 16$

Multiply $24$ by $2$ .

$g (2 x^{2} - 4 x - 3) = 8 x^{4} - 32 x^{3} + 8 x^{2} + 48 x + 2 \cdot 9 - 16$

Multiply $2$ by $9$ .

$g (2 x^{2} - 4 x - 3) = 8 x^{4} - 32 x^{3} + 8 x^{2} + 48 x + 18 - 16$

Step 4

Subtract $16$ from $18$ .

$g (2 x^{2} - 4 x - 3) = 8 x^{4} - 32 x^{3} + 8 x^{2} + 48 x + 2$

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

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