Find f(g(x)) f(x)=3x^2+7x and g(x)=2x^2-x-1

$f (x) = 3 x^{2} + 7 x$ and $g (x) = 2 x^{2} - x - 1$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (2 x^{2} - x - 1)$ by substituting in the value of $g$ into $f$ .

$f (2 x^{2} - x - 1) = 3 {(2 x^{2} - x - 1)}^{2} + 7 (2 x^{2} - x - 1)$

Step 3

Simplify each term.

Rewrite ${(2 x^{2} - x - 1)}^{2}$ as $(2 x^{2} - x - 1) (2 x^{2} - x - 1)$ .

$f (2 x^{2} - x - 1) = 3 ((2 x^{2} - x - 1) (2 x^{2} - x - 1)) + 7 (2 x^{2} - x - 1)$

Expand $(2 x^{2} - x - 1) (2 x^{2} - x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$f (2 x^{2} - x - 1) = 3 (2 x^{2} (2 x^{2}) + 2 x^{2} (- x) + 2 x^{2} \cdot - 1 - x (2 x^{2}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Simplify each term.

Rewrite using the commutative property of multiplication.

$f (2 x^{2} - x - 1) = 3 (2 \cdot (2 x^{2} x^{2}) + 2 x^{2} (- x) + 2 x^{2} \cdot - 1 - x (2 x^{2}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

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

Move $x^{2}$ .

$f (2 x^{2} - x - 1) = 3 (2 \cdot (2 (x^{2} x^{2})) + 2 x^{2} (- x) + 2 x^{2} \cdot - 1 - x (2 x^{2}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

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

$f (2 x^{2} - x - 1) = 3 (2 \cdot (2 x^{2 + 2}) + 2 x^{2} (- x) + 2 x^{2} \cdot - 1 - x (2 x^{2}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Add $2$ and $2$ .

$f (2 x^{2} - x - 1) = 3 (2 \cdot (2 x^{4}) + 2 x^{2} (- x) + 2 x^{2} \cdot - 1 - x (2 x^{2}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Multiply $2$ by $2$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} + 2 x^{2} (- x) + 2 x^{2} \cdot - 1 - x (2 x^{2}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Rewrite using the commutative property of multiplication.

$f (2 x^{2} - x - 1) = 3 (4 x^{4} + 2 \cdot (- 1 x^{2} x) + 2 x^{2} \cdot - 1 - x (2 x^{2}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

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

Move $x$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} + 2 \cdot (- 1 (x \cdot x^{2})) + 2 x^{2} \cdot - 1 - x (2 x^{2}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

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

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

$f (2 x^{2} - x - 1) = 3 (4 x^{4} + 2 \cdot (- 1 (x \cdot x^{2})) + 2 x^{2} \cdot - 1 - x (2 x^{2}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

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

$f (2 x^{2} - x - 1) = 3 (4 x^{4} + 2 \cdot (- 1 x^{1 + 2}) + 2 x^{2} \cdot - 1 - x (2 x^{2}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Add $1$ and $2$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} + 2 \cdot (- 1 x^{3}) + 2 x^{2} \cdot - 1 - x (2 x^{2}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Multiply $2$ by $- 1$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} + 2 x^{2} \cdot - 1 - x (2 x^{2}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Multiply $- 1$ by $2$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - x (2 x^{2}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Rewrite using the commutative property of multiplication.

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 1 \cdot (2 x \cdot x^{2}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

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

Move $x^{2}$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 1 \cdot (2 (x^{2} x)) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

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

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

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 1 \cdot (2 (x^{2} x)) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

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

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 1 \cdot (2 x^{2 + 1}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Add $2$ and $1$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 1 \cdot (2 x^{3}) - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Multiply $- 1$ by $2$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 2 x^{3} - x (- x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Rewrite using the commutative property of multiplication.

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 2 x^{3} - 1 \cdot (- 1 x \cdot x) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

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

Move $x$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 2 x^{3} - 1 \cdot (- 1 (x \cdot x)) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Multiply $x$ by $x$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 2 x^{3} - 1 \cdot (- 1 x^{2}) - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Multiply $- 1$ by $- 1$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 2 x^{3} + 1 x^{2} - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Multiply $x^{2}$ by $1$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 2 x^{3} + x^{2} - x \cdot - 1 - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Multiply $- x \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 2 x^{3} + x^{2} + 1 x - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Multiply $x$ by $1$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 2 x^{3} + x^{2} + x - 1 (2 x^{2}) - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Multiply $2$ by $- 1$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 2 x^{3} + x^{2} + x - 2 x^{2} - 1 (- x) - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Multiply $- 1 (- x)$ .

Multiply $- 1$ by $- 1$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 2 x^{3} + x^{2} + x - 2 x^{2} + 1 x - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Multiply $x$ by $1$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 2 x^{3} + x^{2} + x - 2 x^{2} + x - 1 \cdot - 1) + 7 (2 x^{2} - x - 1)$

Multiply $- 1$ by $- 1$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 2 x^{3} - 2 x^{2} - 2 x^{3} + x^{2} + x - 2 x^{2} + x + 1) + 7 (2 x^{2} - x - 1)$

Subtract $2 x^{3}$ from $- 2 x^{3}$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 4 x^{3} - 2 x^{2} + x^{2} + x - 2 x^{2} + x + 1) + 7 (2 x^{2} - x - 1)$

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

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 4 x^{3} - x^{2} + x - 2 x^{2} + x + 1) + 7 (2 x^{2} - x - 1)$

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

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 4 x^{3} - 3 x^{2} + x + x + 1) + 7 (2 x^{2} - x - 1)$

Add $x$ and $x$ .

$f (2 x^{2} - x - 1) = 3 (4 x^{4} - 4 x^{3} - 3 x^{2} + 2 x + 1) + 7 (2 x^{2} - x - 1)$

Apply the distributive property.

$f (2 x^{2} - x - 1) = 3 (4 x^{4}) + 3 (- 4 x^{3}) + 3 (- 3 x^{2}) + 3 (2 x) + 3 \cdot 1 + 7 (2 x^{2} - x - 1)$

Simplify.

Multiply $4$ by $3$ .

$f (2 x^{2} - x - 1) = 12 x^{4} + 3 (- 4 x^{3}) + 3 (- 3 x^{2}) + 3 (2 x) + 3 \cdot 1 + 7 (2 x^{2} - x - 1)$

Multiply $- 4$ by $3$ .

$f (2 x^{2} - x - 1) = 12 x^{4} - 12 x^{3} + 3 (- 3 x^{2}) + 3 (2 x) + 3 \cdot 1 + 7 (2 x^{2} - x - 1)$

Multiply $- 3$ by $3$ .

$f (2 x^{2} - x - 1) = 12 x^{4} - 12 x^{3} - 9 x^{2} + 3 (2 x) + 3 \cdot 1 + 7 (2 x^{2} - x - 1)$

Multiply $2$ by $3$ .

$f (2 x^{2} - x - 1) = 12 x^{4} - 12 x^{3} - 9 x^{2} + 6 x + 3 \cdot 1 + 7 (2 x^{2} - x - 1)$

Multiply $3$ by $1$ .

$f (2 x^{2} - x - 1) = 12 x^{4} - 12 x^{3} - 9 x^{2} + 6 x + 3 + 7 (2 x^{2} - x - 1)$

Apply the distributive property.

$f (2 x^{2} - x - 1) = 12 x^{4} - 12 x^{3} - 9 x^{2} + 6 x + 3 + 7 (2 x^{2}) + 7 (- x) + 7 \cdot - 1$

Simplify.

Multiply $2$ by $7$ .

$f (2 x^{2} - x - 1) = 12 x^{4} - 12 x^{3} - 9 x^{2} + 6 x + 3 + 14 x^{2} + 7 (- x) + 7 \cdot - 1$

Multiply $- 1$ by $7$ .

$f (2 x^{2} - x - 1) = 12 x^{4} - 12 x^{3} - 9 x^{2} + 6 x + 3 + 14 x^{2} - 7 x + 7 \cdot - 1$

Multiply $7$ by $- 1$ .

$f (2 x^{2} - x - 1) = 12 x^{4} - 12 x^{3} - 9 x^{2} + 6 x + 3 + 14 x^{2} - 7 x - 7$

Step 4

Simplify by adding terms.

Add $- 9 x^{2}$ and $14 x^{2}$ .

$f (2 x^{2} - x - 1) = 12 x^{4} - 12 x^{3} + 5 x^{2} + 6 x + 3 - 7 x - 7$

Subtract $7 x$ from $6 x$ .

$f (2 x^{2} - x - 1) = 12 x^{4} - 12 x^{3} + 5 x^{2} - x + 3 - 7$

Subtract $7$ from $3$ .

$f (2 x^{2} - x - 1) = 12 x^{4} - 12 x^{3} + 5 x^{2} - x - 4$

Find f(g(x)) f(x)=3x^2+7x and g(x)=2x^2-x-1

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