Find f(g(x)) f(x)=9x^2-5x , g(x)=x^2-3x-10

$f (x) = 9 x^{2} - 5 x$ , $g (x) = x^{2} - 3 x - 10$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x^{2} - 3 x - 10)$ by substituting in the value of $g$ into $f$ .

$f (x^{2} - 3 x - 10) = 9 {(x^{2} - 3 x - 10)}^{2} - 5 (x^{2} - 3 x - 10)$

Step 3

Simplify each term.

Rewrite ${(x^{2} - 3 x - 10)}^{2}$ as $(x^{2} - 3 x - 10) (x^{2} - 3 x - 10)$ .

$f (x^{2} - 3 x - 10) = 9 ((x^{2} - 3 x - 10) (x^{2} - 3 x - 10)) - 5 (x^{2} - 3 x - 10)$

Expand $(x^{2} - 3 x - 10) (x^{2} - 3 x - 10)$ by multiplying each term in the first expression by each term in the second expression.

$f (x^{2} - 3 x - 10) = 9 (x^{2} x^{2} + x^{2} (- 3 x) + x^{2} \cdot - 10 - 3 x \cdot x^{2} - 3 x (- 3 x) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

Simplify each term.

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

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

$f (x^{2} - 3 x - 10) = 9 (x^{2 + 2} + x^{2} (- 3 x) + x^{2} \cdot - 10 - 3 x \cdot x^{2} - 3 x (- 3 x) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

Add $2$ and $2$ .

$f (x^{2} - 3 x - 10) = 9 (x^{4} + x^{2} (- 3 x) + x^{2} \cdot - 10 - 3 x \cdot x^{2} - 3 x (- 3 x) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

Rewrite using the commutative property of multiplication.

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{2} x + x^{2} \cdot - 10 - 3 x \cdot x^{2} - 3 x (- 3 x) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

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

Move $x$ .

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 (x \cdot x^{2}) + x^{2} \cdot - 10 - 3 x \cdot x^{2} - 3 x (- 3 x) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

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

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

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 (x \cdot x^{2}) + x^{2} \cdot - 10 - 3 x \cdot x^{2} - 3 x (- 3 x) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

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

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{1 + 2} + x^{2} \cdot - 10 - 3 x \cdot x^{2} - 3 x (- 3 x) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

Add $1$ and $2$ .

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{3} + x^{2} \cdot - 10 - 3 x \cdot x^{2} - 3 x (- 3 x) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

Move $- 10$ to the left of $x^{2}$ .

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{3} - 10 \cdot x^{2} - 3 x \cdot x^{2} - 3 x (- 3 x) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

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

Move $x^{2}$ .

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{3} - 10 x^{2} - 3 (x^{2} x) - 3 x (- 3 x) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

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

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

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{3} - 10 x^{2} - 3 (x^{2} x) - 3 x (- 3 x) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

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

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{3} - 10 x^{2} - 3 x^{2 + 1} - 3 x (- 3 x) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

Add $2$ and $1$ .

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{3} - 10 x^{2} - 3 x^{3} - 3 x (- 3 x) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

Rewrite using the commutative property of multiplication.

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{3} - 10 x^{2} - 3 x^{3} - 3 \cdot (- 3 x \cdot x) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

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

Move $x$ .

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{3} - 10 x^{2} - 3 x^{3} - 3 \cdot (- 3 (x \cdot x)) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

Multiply $x$ by $x$ .

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{3} - 10 x^{2} - 3 x^{3} - 3 \cdot (- 3 x^{2}) - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

Multiply $- 3$ by $- 3$ .

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{3} - 10 x^{2} - 3 x^{3} + 9 x^{2} - 3 x \cdot - 10 - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

Multiply $- 10$ by $- 3$ .

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{3} - 10 x^{2} - 3 x^{3} + 9 x^{2} + 30 x - 10 x^{2} - 10 (- 3 x) - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

Multiply $- 3$ by $- 10$ .

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{3} - 10 x^{2} - 3 x^{3} + 9 x^{2} + 30 x - 10 x^{2} + 30 x - 10 \cdot - 10) - 5 (x^{2} - 3 x - 10)$

Multiply $- 10$ by $- 10$ .

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 3 x^{3} - 10 x^{2} - 3 x^{3} + 9 x^{2} + 30 x - 10 x^{2} + 30 x + 100) - 5 (x^{2} - 3 x - 10)$

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

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 6 x^{3} - 10 x^{2} + 9 x^{2} + 30 x - 10 x^{2} + 30 x + 100) - 5 (x^{2} - 3 x - 10)$

Add $- 10 x^{2}$ and $9 x^{2}$ .

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 6 x^{3} - x^{2} + 30 x - 10 x^{2} + 30 x + 100) - 5 (x^{2} - 3 x - 10)$

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

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 6 x^{3} - 11 x^{2} + 30 x + 30 x + 100) - 5 (x^{2} - 3 x - 10)$

Add $30 x$ and $30 x$ .

$f (x^{2} - 3 x - 10) = 9 (x^{4} - 6 x^{3} - 11 x^{2} + 60 x + 100) - 5 (x^{2} - 3 x - 10)$

Apply the distributive property.

$f (x^{2} - 3 x - 10) = 9 x^{4} + 9 (- 6 x^{3}) + 9 (- 11 x^{2}) + 9 (60 x) + 9 \cdot 100 - 5 (x^{2} - 3 x - 10)$

Simplify.

Multiply $- 6$ by $9$ .

$f (x^{2} - 3 x - 10) = 9 x^{4} - 54 x^{3} + 9 (- 11 x^{2}) + 9 (60 x) + 9 \cdot 100 - 5 (x^{2} - 3 x - 10)$

Multiply $- 11$ by $9$ .

$f (x^{2} - 3 x - 10) = 9 x^{4} - 54 x^{3} - 99 x^{2} + 9 (60 x) + 9 \cdot 100 - 5 (x^{2} - 3 x - 10)$

Multiply $60$ by $9$ .

$f (x^{2} - 3 x - 10) = 9 x^{4} - 54 x^{3} - 99 x^{2} + 540 x + 9 \cdot 100 - 5 (x^{2} - 3 x - 10)$

Multiply $9$ by $100$ .

$f (x^{2} - 3 x - 10) = 9 x^{4} - 54 x^{3} - 99 x^{2} + 540 x + 900 - 5 (x^{2} - 3 x - 10)$

Apply the distributive property.

$f (x^{2} - 3 x - 10) = 9 x^{4} - 54 x^{3} - 99 x^{2} + 540 x + 900 - 5 x^{2} - 5 (- 3 x) - 5 \cdot - 10$

Simplify.

Multiply $- 3$ by $- 5$ .

$f (x^{2} - 3 x - 10) = 9 x^{4} - 54 x^{3} - 99 x^{2} + 540 x + 900 - 5 x^{2} + 15 x - 5 \cdot - 10$

Multiply $- 5$ by $- 10$ .

$f (x^{2} - 3 x - 10) = 9 x^{4} - 54 x^{3} - 99 x^{2} + 540 x + 900 - 5 x^{2} + 15 x + 50$

Step 4

Simplify by adding terms.

Subtract $5 x^{2}$ from $- 99 x^{2}$ .

$f (x^{2} - 3 x - 10) = 9 x^{4} - 54 x^{3} - 104 x^{2} + 540 x + 900 + 15 x + 50$

Add $540 x$ and $15 x$ .

$f (x^{2} - 3 x - 10) = 9 x^{4} - 54 x^{3} - 104 x^{2} + 555 x + 900 + 50$

Add $900$ and $50$ .

$f (x^{2} - 3 x - 10) = 9 x^{4} - 54 x^{3} - 104 x^{2} + 555 x + 950$

Find f(g(x)) f(x)=9x^2-5x , g(x)=x^2-3x-10

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