Solve the Function Operation f(x)=x^2+3x , g(x)=5x-4 , (g*f)(3)=

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

Step 1

Evaluate $g \cdot f$ .

Replace the function designators in $g \cdot f$ with the actual functions.

$(5 x - 4) \cdot (x^{2} + 3 x)$

Expand $(5 x - 4) (x^{2} + 3 x)$ using the FOIL Method.

Apply the distributive property.

$5 x (x^{2} + 3 x) - 4 (x^{2} + 3 x)$

Apply the distributive property.

$5 x \cdot x^{2} + 5 x (3 x) - 4 (x^{2} + 3 x)$

Apply the distributive property.

$5 x \cdot x^{2} + 5 x (3 x) - 4 x^{2} - 4 (3 x)$

Simplify and combine like terms.

Simplify each term.

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

Move $x^{2}$ .

$5 (x^{2} x) + 5 x (3 x) - 4 x^{2} - 4 (3 x)$

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

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

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

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

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

Add $2$ and $1$ .

$5 x^{3} + 5 x (3 x) - 4 x^{2} - 4 (3 x)$

Rewrite using the commutative property of multiplication.

$5 x^{3} + 5 \cdot 3 x \cdot x - 4 x^{2} - 4 (3 x)$

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

Move $x$ .

$5 x^{3} + 5 \cdot 3 (x \cdot x) - 4 x^{2} - 4 (3 x)$

Multiply $x$ by $x$ .

$5 x^{3} + 5 \cdot 3 x^{2} - 4 x^{2} - 4 (3 x)$

Multiply $5$ by $3$ .

$5 x^{3} + 15 x^{2} - 4 x^{2} - 4 (3 x)$

Multiply $3$ by $- 4$ .

$5 x^{3} + 15 x^{2} - 4 x^{2} - 12 x$

Subtract $4 x^{2}$ from $15 x^{2}$ .

$5 x^{3} + 11 x^{2} - 12 x$

Step 2

Evaluate $g \cdot f$ at $3$ .

$5 {(3)}^{3} + 11 {(3)}^{2} - 12 \cdot 3$

Step 3

Simplify $5 {(3)}^{3} + 11 {(3)}^{2} - 12 \cdot 3$ .

Remove parentheses.

$5 {(3)}^{3} + 11 {(3)}^{2} - 12 \cdot 3$

Simplify each term.

Raise $3$ to the power of $3$ .

$5 \cdot 27 + 11 {(3)}^{2} - 12 \cdot 3$

Multiply $5$ by $27$ .

$135 + 11 {(3)}^{2} - 12 \cdot 3$

Raise $3$ to the power of $2$ .

$135 + 11 \cdot 9 - 12 \cdot 3$

Multiply $11$ by $9$ .

$135 + 99 - 12 \cdot 3$

Multiply $- 12$ by $3$ .

$135 + 99 - 36$

Simplify by adding and subtracting.

Add $135$ and $99$ .

$234 - 36$

Subtract $36$ from $234$ .

$198$

Solve the Function Operation f(x)=x^2+3x , g(x)=5x-4 , (g*f)(3)=

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