Find the Domain of the Product f(x)=x^2-49 , g(x)=2x+3

$f (x) = x^{2} - 49$ , $g (x) = 2 x + 3$

Find the product of the functions.

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

$(x^{2} - 49) \cdot (2 x + 3)$

Simplify.

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Expand $(x^{2} - 49) (2 x + 3)$ using the FOIL Method.

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Apply the distributive property.

$x^{2} (2 x + 3) - 49 (2 x + 3)$

Apply the distributive property.

$x^{2} (2 x) + x^{2} \cdot 3 - 49 (2 x + 3)$

Apply the distributive property.

$x^{2} (2 x) + x^{2} \cdot 3 - 49 (2 x) - 49 \cdot 3$

Simplify each term.

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Rewrite using the commutative property of multiplication.

$2 x^{2} x + x^{2} \cdot 3 - 49 (2 x) - 49 \cdot 3$

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

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Move $x$ .

$2 (x \cdot x^{2}) + x^{2} \cdot 3 - 49 (2 x) - 49 \cdot 3$

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

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

$2 (x^{1} x^{2}) + x^{2} \cdot 3 - 49 (2 x) - 49 \cdot 3$

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

$2 x^{1 + 2} + x^{2} \cdot 3 - 49 (2 x) - 49 \cdot 3$

Add $1$ and $2$ .

$2 x^{3} + x^{2} \cdot 3 - 49 (2 x) - 49 \cdot 3$

Move $3$ to the left of $x^{2}$ .

$2 x^{3} + 3 \cdot x^{2} - 49 (2 x) - 49 \cdot 3$

Multiply $2$ by $- 49$ .

$2 x^{3} + 3 x^{2} - 98 x - 49 \cdot 3$

Multiply $- 49$ by $3$ .

$2 x^{3} + 3 x^{2} - 98 x - 147$

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:

${x | x \in ℝ}$

Find the Domain of the Product f(x)=x^2-49 , g(x)=2x+3

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