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

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

Step 1

Find the product of the functions.

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

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

Simplify.

Apply the distributive property.

$\sqrt{x + 7} x^{2} + \sqrt{x + 7} \cdot - 7$

Move $- 7$ to the left of $\sqrt{x + 7}$ .

$\sqrt{x + 7} x^{2} - 7 \sqrt{x + 7}$

Step 2

Set the radicand in $\sqrt{x + 7}$ greater than or equal to $0$ to find where the expression is defined.

$x + 7 \geq 0$

Step 3

Subtract $7$ from both sides of the inequality.

$x \geq - 7$

Step 4

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- 7, \infty)$

Set-Builder Notation:

${x | x \geq - 7}$

Step 5