Find the Domain of the Product of the Functions f(x)=7x+11 g(x)=-2x+3 h(x)=3x^2+1

$f (x) = 7 x + 11$ $g (x) = - 2 x + 3$ $h (x) = 3 x^{2} + 1$

Step 1

Find the product of the functions.

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

$(7 x + 11) \cdot (- 2 x + 3) \cdot (3 x^{2} + 1)$

Simplify.

Expand $(7 x + 11) (- 2 x + 3)$ using the FOIL Method.

Apply the distributive property.

$(7 x (- 2 x + 3) + 11 (- 2 x + 3)) \cdot (3 x^{2} + 1)$

Apply the distributive property.

$(7 x (- 2 x) + 7 x \cdot 3 + 11 (- 2 x + 3)) \cdot (3 x^{2} + 1)$

Apply the distributive property.

$(7 x (- 2 x) + 7 x \cdot 3 + 11 (- 2 x) + 11 \cdot 3) \cdot (3 x^{2} + 1)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$(7 \cdot - 2 x \cdot x + 7 x \cdot 3 + 11 (- 2 x) + 11 \cdot 3) \cdot (3 x^{2} + 1)$

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

Move $x$ .

$(7 \cdot - 2 (x \cdot x) + 7 x \cdot 3 + 11 (- 2 x) + 11 \cdot 3) \cdot (3 x^{2} + 1)$

Multiply $x$ by $x$ .

$(7 \cdot - 2 x^{2} + 7 x \cdot 3 + 11 (- 2 x) + 11 \cdot 3) \cdot (3 x^{2} + 1)$

Multiply $7$ by $- 2$ .

$(- 14 x^{2} + 7 x \cdot 3 + 11 (- 2 x) + 11 \cdot 3) \cdot (3 x^{2} + 1)$

Multiply $3$ by $7$ .

$(- 14 x^{2} + 21 x + 11 (- 2 x) + 11 \cdot 3) \cdot (3 x^{2} + 1)$

Multiply $- 2$ by $11$ .

$(- 14 x^{2} + 21 x - 22 x + 11 \cdot 3) \cdot (3 x^{2} + 1)$

Multiply $11$ by $3$ .

$(- 14 x^{2} + 21 x - 22 x + 33) \cdot (3 x^{2} + 1)$

Subtract $22 x$ from $21 x$ .

$(- 14 x^{2} - x + 33) \cdot (3 x^{2} + 1)$

Expand $(- 14 x^{2} - x + 33) (3 x^{2} + 1)$ by multiplying each term in the first expression by each term in the second expression.

$- 14 x^{2} (3 x^{2}) - 14 x^{2} \cdot 1 - x (3 x^{2}) - x \cdot 1 + 33 (3 x^{2}) + 33 \cdot 1$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 14 \cdot 3 x^{2} x^{2} - 14 x^{2} \cdot 1 - x (3 x^{2}) - x \cdot 1 + 33 (3 x^{2}) + 33 \cdot 1$

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

Move $x^{2}$ .

$- 14 \cdot 3 (x^{2} x^{2}) - 14 x^{2} \cdot 1 - x (3 x^{2}) - x \cdot 1 + 33 (3 x^{2}) + 33 \cdot 1$

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

$- 14 \cdot 3 x^{2 + 2} - 14 x^{2} \cdot 1 - x (3 x^{2}) - x \cdot 1 + 33 (3 x^{2}) + 33 \cdot 1$

Add $2$ and $2$ .

$- 14 \cdot 3 x^{4} - 14 x^{2} \cdot 1 - x (3 x^{2}) - x \cdot 1 + 33 (3 x^{2}) + 33 \cdot 1$

Multiply $- 14$ by $3$ .

$- 42 x^{4} - 14 x^{2} \cdot 1 - x (3 x^{2}) - x \cdot 1 + 33 (3 x^{2}) + 33 \cdot 1$

Multiply $- 14$ by $1$ .

$- 42 x^{4} - 14 x^{2} - x (3 x^{2}) - x \cdot 1 + 33 (3 x^{2}) + 33 \cdot 1$

Rewrite using the commutative property of multiplication.

$- 42 x^{4} - 14 x^{2} - 1 \cdot 3 x \cdot x^{2} - x \cdot 1 + 33 (3 x^{2}) + 33 \cdot 1$

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

Move $x^{2}$ .

$- 42 x^{4} - 14 x^{2} - 1 \cdot 3 (x^{2} x) - x \cdot 1 + 33 (3 x^{2}) + 33 \cdot 1$

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

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

$- 42 x^{4} - 14 x^{2} - 1 \cdot 3 (x^{2} x^{1}) - x \cdot 1 + 33 (3 x^{2}) + 33 \cdot 1$

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

$- 42 x^{4} - 14 x^{2} - 1 \cdot 3 x^{2 + 1} - x \cdot 1 + 33 (3 x^{2}) + 33 \cdot 1$

Add $2$ and $1$ .

$- 42 x^{4} - 14 x^{2} - 1 \cdot 3 x^{3} - x \cdot 1 + 33 (3 x^{2}) + 33 \cdot 1$

Multiply $- 1$ by $3$ .

$- 42 x^{4} - 14 x^{2} - 3 x^{3} - x \cdot 1 + 33 (3 x^{2}) + 33 \cdot 1$

Multiply $- 1$ by $1$ .

$- 42 x^{4} - 14 x^{2} - 3 x^{3} - x + 33 (3 x^{2}) + 33 \cdot 1$

Multiply $3$ by $33$ .

$- 42 x^{4} - 14 x^{2} - 3 x^{3} - x + 99 x^{2} + 33 \cdot 1$

Multiply $33$ by $1$ .

$- 42 x^{4} - 14 x^{2} - 3 x^{3} - x + 99 x^{2} + 33$

Add $- 14 x^{2}$ and $99 x^{2}$ .

$- 42 x^{4} + 85 x^{2} - 3 x^{3} - x + 33$

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)=7x+11 g(x)=-2x+3 h(x)=3x^2+1

Related Questions

Support LCMGCF.com by disabling your adblocker.