Find the Domain of the Product of the Functions f(x)=x+3 , g(x)=2x^2-5x-3

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

Step 1

Find the product of the functions.

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

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

Simplify.

Expand $(x + 3) (2 x^{2} - 5 x - 3)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Move $- 3$ to the left of $x$ .

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

Multiply $2$ by $3$ .

$2 x^{3} - 5 x^{2} - 3 x + 6 x^{2} + 3 (- 5 x) + 3 \cdot - 3$

Multiply $- 5$ by $3$ .

$2 x^{3} - 5 x^{2} - 3 x + 6 x^{2} - 15 x + 3 \cdot - 3$

Multiply $3$ by $- 3$ .

$2 x^{3} - 5 x^{2} - 3 x + 6 x^{2} - 15 x - 9$

Simplify by adding terms.

Add $- 5 x^{2}$ and $6 x^{2}$ .

$2 x^{3} + x^{2} - 3 x - 15 x - 9$

Subtract $15 x$ from $- 3 x$ .

$2 x^{3} + x^{2} - 18 x - 9$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

Find the Domain of the Product of the Functions f(x)=x+3 , g(x)=2x^2-5x-3

Related Questions

Support LCMGCF.com by disabling your adblocker.