Find the Domain of the Product of the Functions f(x)=6x-12 g(x)=5x+1 h(x)=x^2-4

$f (x) = 6 x - 12$ $g (x) = 5 x + 1$ $h (x) = x^{2} - 4$

Step 1

Find the product of the functions.

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

$(6 x - 12) \cdot (5 x + 1) \cdot (x^{2} - 4)$

Simplify.

Expand $(6 x - 12) (5 x + 1)$ using the FOIL Method.

Apply the distributive property.

$(6 x (5 x + 1) - 12 (5 x + 1)) \cdot (x^{2} - 4)$

Apply the distributive property.

$(6 x (5 x) + 6 x \cdot 1 - 12 (5 x + 1)) \cdot (x^{2} - 4)$

Apply the distributive property.

$(6 x (5 x) + 6 x \cdot 1 - 12 (5 x) - 12 \cdot 1) \cdot (x^{2} - 4)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$(6 \cdot 5 x \cdot x + 6 x \cdot 1 - 12 (5 x) - 12 \cdot 1) \cdot (x^{2} - 4)$

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

Move $x$ .

$(6 \cdot 5 (x \cdot x) + 6 x \cdot 1 - 12 (5 x) - 12 \cdot 1) \cdot (x^{2} - 4)$

Multiply $x$ by $x$ .

$(6 \cdot 5 x^{2} + 6 x \cdot 1 - 12 (5 x) - 12 \cdot 1) \cdot (x^{2} - 4)$

Multiply $6$ by $5$ .

$(30 x^{2} + 6 x \cdot 1 - 12 (5 x) - 12 \cdot 1) \cdot (x^{2} - 4)$

Multiply $6$ by $1$ .

$(30 x^{2} + 6 x - 12 (5 x) - 12 \cdot 1) \cdot (x^{2} - 4)$

Multiply $5$ by $- 12$ .

$(30 x^{2} + 6 x - 60 x - 12 \cdot 1) \cdot (x^{2} - 4)$

Multiply $- 12$ by $1$ .

$(30 x^{2} + 6 x - 60 x - 12) \cdot (x^{2} - 4)$

Subtract $60 x$ from $6 x$ .

$(30 x^{2} - 54 x - 12) \cdot (x^{2} - 4)$

Expand $(30 x^{2} - 54 x - 12) (x^{2} - 4)$ by multiplying each term in the first expression by each term in the second expression.

$30 x^{2} x^{2} + 30 x^{2} \cdot - 4 - 54 x \cdot x^{2} - 54 x \cdot - 4 - 12 x^{2} - 12 \cdot - 4$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$30 (x^{2} x^{2}) + 30 x^{2} \cdot - 4 - 54 x \cdot x^{2} - 54 x \cdot - 4 - 12 x^{2} - 12 \cdot - 4$

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

$30 x^{2 + 2} + 30 x^{2} \cdot - 4 - 54 x \cdot x^{2} - 54 x \cdot - 4 - 12 x^{2} - 12 \cdot - 4$

Add $2$ and $2$ .

$30 x^{4} + 30 x^{2} \cdot - 4 - 54 x \cdot x^{2} - 54 x \cdot - 4 - 12 x^{2} - 12 \cdot - 4$

Multiply $- 4$ by $30$ .

$30 x^{4} - 120 x^{2} - 54 x \cdot x^{2} - 54 x \cdot - 4 - 12 x^{2} - 12 \cdot - 4$

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

Move $x^{2}$ .

$30 x^{4} - 120 x^{2} - 54 (x^{2} x) - 54 x \cdot - 4 - 12 x^{2} - 12 \cdot - 4$

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

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

$30 x^{4} - 120 x^{2} - 54 (x^{2} x^{1}) - 54 x \cdot - 4 - 12 x^{2} - 12 \cdot - 4$

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

$30 x^{4} - 120 x^{2} - 54 x^{2 + 1} - 54 x \cdot - 4 - 12 x^{2} - 12 \cdot - 4$

Add $2$ and $1$ .

$30 x^{4} - 120 x^{2} - 54 x^{3} - 54 x \cdot - 4 - 12 x^{2} - 12 \cdot - 4$

Multiply $- 4$ by $- 54$ .

$30 x^{4} - 120 x^{2} - 54 x^{3} + 216 x - 12 x^{2} - 12 \cdot - 4$

Multiply $- 12$ by $- 4$ .

$30 x^{4} - 120 x^{2} - 54 x^{3} + 216 x - 12 x^{2} + 48$

Subtract $12 x^{2}$ from $- 120 x^{2}$ .

$30 x^{4} - 132 x^{2} - 54 x^{3} + 216 x + 48$

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)=6x-12 g(x)=5x+1 h(x)=x^2-4

Related Questions

Support LCMGCF.com by disabling your adblocker.