Find the Domain of the Product of the Functions f(x)=2x^2+x-3 and g(x)=x-1

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

Step 1

Find the product of the functions.

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

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

Simplify.

Expand $(2 x^{2} + x - 3) (x - 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify terms.

Simplify each term.

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $x$ by $x$ .

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

Move $- 1$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $- 3$ by $- 1$ .

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

Simplify by adding terms.

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

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

Subtract $3 x$ from $- x$ .

$2 x^{3} - x^{2} - 4 x + 3$

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

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