Find the Domain of the Product of the Functions f(x)=x+4 , g(x)=2x^2-7x-4

$f (x) = x + 4$ , $g (x) = 2 x^{2} - 7 x - 4$

Step 1

Find the product of the functions.

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

$(x + 4) \cdot (2 x^{2} - 7 x - 4)$

Simplify.

Expand $(x + 4) (2 x^{2} - 7 x - 4)$ by multiplying each term in the first expression by each term in the second expression.

$x (2 x^{2}) + x (- 7 x) + x \cdot - 4 + 4 (2 x^{2}) + 4 (- 7 x) + 4 \cdot - 4$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 x \cdot x^{2} + x (- 7 x) + x \cdot - 4 + 4 (2 x^{2}) + 4 (- 7 x) + 4 \cdot - 4$

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

Move $x^{2}$ .

$2 (x^{2} x) + x (- 7 x) + x \cdot - 4 + 4 (2 x^{2}) + 4 (- 7 x) + 4 \cdot - 4$

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

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

$2 (x^{2} x^{1}) + x (- 7 x) + x \cdot - 4 + 4 (2 x^{2}) + 4 (- 7 x) + 4 \cdot - 4$

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

$2 x^{2 + 1} + x (- 7 x) + x \cdot - 4 + 4 (2 x^{2}) + 4 (- 7 x) + 4 \cdot - 4$

Add $2$ and $1$ .

$2 x^{3} + x (- 7 x) + x \cdot - 4 + 4 (2 x^{2}) + 4 (- 7 x) + 4 \cdot - 4$

Rewrite using the commutative property of multiplication.

$2 x^{3} - 7 x \cdot x + x \cdot - 4 + 4 (2 x^{2}) + 4 (- 7 x) + 4 \cdot - 4$

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

Move $x$ .

$2 x^{3} - 7 (x \cdot x) + x \cdot - 4 + 4 (2 x^{2}) + 4 (- 7 x) + 4 \cdot - 4$

Multiply $x$ by $x$ .

$2 x^{3} - 7 x^{2} + x \cdot - 4 + 4 (2 x^{2}) + 4 (- 7 x) + 4 \cdot - 4$

Move $- 4$ to the left of $x$ .

$2 x^{3} - 7 x^{2} - 4 \cdot x + 4 (2 x^{2}) + 4 (- 7 x) + 4 \cdot - 4$

Multiply $2$ by $4$ .

$2 x^{3} - 7 x^{2} - 4 x + 8 x^{2} + 4 (- 7 x) + 4 \cdot - 4$

Multiply $- 7$ by $4$ .

$2 x^{3} - 7 x^{2} - 4 x + 8 x^{2} - 28 x + 4 \cdot - 4$

Multiply $4$ by $- 4$ .

$2 x^{3} - 7 x^{2} - 4 x + 8 x^{2} - 28 x - 16$

Simplify by adding terms.

Add $- 7 x^{2}$ and $8 x^{2}$ .

$2 x^{3} + x^{2} - 4 x - 28 x - 16$

Subtract $28 x$ from $- 4 x$ .

$2 x^{3} + x^{2} - 32 x - 16$

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+4 , g(x)=2x^2-7x-4

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