Find f(g(x)) f(x)=3x^2+2x+5 , g(x)=9x^2+4x+9

$f (x) = 3 x^{2} + 2 x + 5$ , $g (x) = 9 x^{2} + 4 x + 9$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (9 x^{2} + 4 x + 9)$ by substituting in the value of $g$ into $f$ .

$f (9 x^{2} + 4 x + 9) = 3 {(9 x^{2} + 4 x + 9)}^{2} + 2 (9 x^{2} + 4 x + 9) + 5$

Step 3

Simplify each term.

Rewrite ${(9 x^{2} + 4 x + 9)}^{2}$ as $(9 x^{2} + 4 x + 9) (9 x^{2} + 4 x + 9)$ .

$f (9 x^{2} + 4 x + 9) = 3 ((9 x^{2} + 4 x + 9) (9 x^{2} + 4 x + 9)) + 2 (9 x^{2} + 4 x + 9) + 5$

Expand $(9 x^{2} + 4 x + 9) (9 x^{2} + 4 x + 9)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Multiply $9$ by $9$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 9 x^{2} (4 x) + 9 x^{2} \cdot 9 + 4 x (9 x^{2}) + 4 x (4 x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

Rewrite using the commutative property of multiplication.

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 9 \cdot (4 x^{2} x) + 9 x^{2} \cdot 9 + 4 x (9 x^{2}) + 4 x (4 x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

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

Move $x$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 9 \cdot (4 (x \cdot x^{2})) + 9 x^{2} \cdot 9 + 4 x (9 x^{2}) + 4 x (4 x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

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

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

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 9 \cdot (4 (x \cdot x^{2})) + 9 x^{2} \cdot 9 + 4 x (9 x^{2}) + 4 x (4 x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

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

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 9 \cdot (4 x^{1 + 2}) + 9 x^{2} \cdot 9 + 4 x (9 x^{2}) + 4 x (4 x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

Add $1$ and $2$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 9 \cdot (4 x^{3}) + 9 x^{2} \cdot 9 + 4 x (9 x^{2}) + 4 x (4 x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

Multiply $9$ by $4$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 9 x^{2} \cdot 9 + 4 x (9 x^{2}) + 4 x (4 x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

Multiply $9$ by $9$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 4 x (9 x^{2}) + 4 x (4 x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

Rewrite using the commutative property of multiplication.

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 4 \cdot (9 x \cdot x^{2}) + 4 x (4 x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

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

Move $x^{2}$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 4 \cdot (9 (x^{2} x)) + 4 x (4 x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

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

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

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 4 \cdot (9 (x^{2} x)) + 4 x (4 x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

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

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 4 \cdot (9 x^{2 + 1}) + 4 x (4 x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

Add $2$ and $1$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 4 \cdot (9 x^{3}) + 4 x (4 x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

Multiply $4$ by $9$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 36 x^{3} + 4 x (4 x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

Rewrite using the commutative property of multiplication.

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 36 x^{3} + 4 \cdot (4 x \cdot x) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

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

Move $x$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 36 x^{3} + 4 \cdot (4 (x \cdot x)) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

Multiply $x$ by $x$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 36 x^{3} + 4 \cdot (4 x^{2}) + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

Multiply $4$ by $4$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 36 x^{3} + 16 x^{2} + 4 x \cdot 9 + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

Multiply $9$ by $4$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 36 x^{3} + 16 x^{2} + 36 x + 9 (9 x^{2}) + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

Multiply $9$ by $9$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 36 x^{3} + 16 x^{2} + 36 x + 81 x^{2} + 9 (4 x) + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

Multiply $4$ by $9$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 36 x^{3} + 16 x^{2} + 36 x + 81 x^{2} + 36 x + 9 \cdot 9) + 2 (9 x^{2} + 4 x + 9) + 5$

Multiply $9$ by $9$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 36 x^{3} + 81 x^{2} + 36 x^{3} + 16 x^{2} + 36 x + 81 x^{2} + 36 x + 81) + 2 (9 x^{2} + 4 x + 9) + 5$

Add $36 x^{3}$ and $36 x^{3}$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 72 x^{3} + 81 x^{2} + 16 x^{2} + 36 x + 81 x^{2} + 36 x + 81) + 2 (9 x^{2} + 4 x + 9) + 5$

Add $81 x^{2}$ and $16 x^{2}$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 72 x^{3} + 97 x^{2} + 36 x + 81 x^{2} + 36 x + 81) + 2 (9 x^{2} + 4 x + 9) + 5$

Add $97 x^{2}$ and $81 x^{2}$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 72 x^{3} + 178 x^{2} + 36 x + 36 x + 81) + 2 (9 x^{2} + 4 x + 9) + 5$

Add $36 x$ and $36 x$ .

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4} + 72 x^{3} + 178 x^{2} + 72 x + 81) + 2 (9 x^{2} + 4 x + 9) + 5$

Apply the distributive property.

$f (9 x^{2} + 4 x + 9) = 3 (81 x^{4}) + 3 (72 x^{3}) + 3 (178 x^{2}) + 3 (72 x) + 3 \cdot 81 + 2 (9 x^{2} + 4 x + 9) + 5$

Simplify.

Multiply $81$ by $3$ .

$f (9 x^{2} + 4 x + 9) = 243 x^{4} + 3 (72 x^{3}) + 3 (178 x^{2}) + 3 (72 x) + 3 \cdot 81 + 2 (9 x^{2} + 4 x + 9) + 5$

Multiply $72$ by $3$ .

$f (9 x^{2} + 4 x + 9) = 243 x^{4} + 216 x^{3} + 3 (178 x^{2}) + 3 (72 x) + 3 \cdot 81 + 2 (9 x^{2} + 4 x + 9) + 5$

Multiply $178$ by $3$ .

$f (9 x^{2} + 4 x + 9) = 243 x^{4} + 216 x^{3} + 534 x^{2} + 3 (72 x) + 3 \cdot 81 + 2 (9 x^{2} + 4 x + 9) + 5$

Multiply $72$ by $3$ .

$f (9 x^{2} + 4 x + 9) = 243 x^{4} + 216 x^{3} + 534 x^{2} + 216 x + 3 \cdot 81 + 2 (9 x^{2} + 4 x + 9) + 5$

Multiply $3$ by $81$ .

$f (9 x^{2} + 4 x + 9) = 243 x^{4} + 216 x^{3} + 534 x^{2} + 216 x + 243 + 2 (9 x^{2} + 4 x + 9) + 5$

Apply the distributive property.

$f (9 x^{2} + 4 x + 9) = 243 x^{4} + 216 x^{3} + 534 x^{2} + 216 x + 243 + 2 (9 x^{2}) + 2 (4 x) + 2 \cdot 9 + 5$

Simplify.

Multiply $9$ by $2$ .

$f (9 x^{2} + 4 x + 9) = 243 x^{4} + 216 x^{3} + 534 x^{2} + 216 x + 243 + 18 x^{2} + 2 (4 x) + 2 \cdot 9 + 5$

Multiply $4$ by $2$ .

$f (9 x^{2} + 4 x + 9) = 243 x^{4} + 216 x^{3} + 534 x^{2} + 216 x + 243 + 18 x^{2} + 8 x + 2 \cdot 9 + 5$

Multiply $2$ by $9$ .

$f (9 x^{2} + 4 x + 9) = 243 x^{4} + 216 x^{3} + 534 x^{2} + 216 x + 243 + 18 x^{2} + 8 x + 18 + 5$

Step 4

Simplify by adding terms.

Add $534 x^{2}$ and $18 x^{2}$ .

$f (9 x^{2} + 4 x + 9) = 243 x^{4} + 216 x^{3} + 552 x^{2} + 216 x + 243 + 8 x + 18 + 5$

Add $216 x$ and $8 x$ .

$f (9 x^{2} + 4 x + 9) = 243 x^{4} + 216 x^{3} + 552 x^{2} + 224 x + 243 + 18 + 5$

Simplify by adding numbers.

Add $243$ and $18$ .

$f (9 x^{2} + 4 x + 9) = 243 x^{4} + 216 x^{3} + 552 x^{2} + 224 x + 261 + 5$

Add $261$ and $5$ .

$f (9 x^{2} + 4 x + 9) = 243 x^{4} + 216 x^{3} + 552 x^{2} + 224 x + 266$

Find f(g(x)) f(x)=3x^2+2x+5 , g(x)=9x^2+4x+9

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