Find g(f(x)) f(x)=3x^2-8x , g(x)=x^2-8x+16

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

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (3 x^{2} - 8 x)$ by substituting in the value of $f$ into $g$ .

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

Step 3

Simplify each term.

Rewrite ${(3 x^{2} - 8 x)}^{2}$ as $(3 x^{2} - 8 x) (3 x^{2} - 8 x)$ .

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

Expand $(3 x^{2} - 8 x) (3 x^{2} - 8 x)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Multiply $3$ by $3$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

$g (3 x^{2} - 8 x) = 9 x^{4} + 3 \cdot (- 8 x^{1 + 2}) - 8 x (3 x^{2}) - 8 x (- 8 x) - 8 (3 x^{2} - 8 x) + 16$

Add $1$ and $2$ .

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

Multiply $3$ by $- 8$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

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

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

$g (3 x^{2} - 8 x) = 9 x^{4} - 24 x^{3} - 8 \cdot (3 x^{2 + 1}) - 8 x (- 8 x) - 8 (3 x^{2} - 8 x) + 16$

Add $2$ and $1$ .

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

Multiply $- 8$ by $3$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 8$ by $- 8$ .

$g (3 x^{2} - 8 x) = 9 x^{4} - 24 x^{3} - 24 x^{3} + 64 x^{2} - 8 (3 x^{2} - 8 x) + 16$

Subtract $24 x^{3}$ from $- 24 x^{3}$ .

$g (3 x^{2} - 8 x) = 9 x^{4} - 48 x^{3} + 64 x^{2} - 8 (3 x^{2} - 8 x) + 16$

Apply the distributive property.

$g (3 x^{2} - 8 x) = 9 x^{4} - 48 x^{3} + 64 x^{2} - 8 (3 x^{2}) - 8 (- 8 x) + 16$

Multiply $3$ by $- 8$ .

$g (3 x^{2} - 8 x) = 9 x^{4} - 48 x^{3} + 64 x^{2} - 24 x^{2} - 8 (- 8 x) + 16$

Multiply $- 8$ by $- 8$ .

$g (3 x^{2} - 8 x) = 9 x^{4} - 48 x^{3} + 64 x^{2} - 24 x^{2} + 64 x + 16$

Step 4

Subtract $24 x^{2}$ from $64 x^{2}$ .

$g (3 x^{2} - 8 x) = 9 x^{4} - 48 x^{3} + 40 x^{2} + 64 x + 16$

Find g(f(x)) f(x)=3x^2-8x , g(x)=x^2-8x+16

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