Find the Domain of the Quotient of the Two Functions f(x)=x-8 , g(x)=4x^2

$f (x) = x - 8$ , $g (x) = 4 x^{2}$

Step 1

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

$\frac{x - 8}{4 x^{2}}$

Step 2

Set the denominator in $\frac{x - 8}{4 x^{2}}$ equal to $0$ to find where the expression is undefined.

$4 x^{2} = 0$

Step 3

Solve for $x$ .

Divide each term in $4 x^{2} = 0$ by $4$ and simplify.

Divide each term in $4 x^{2} = 0$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{0}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{0}{4}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{4}$

Simplify the right side.

Divide $0$ by $4$ .

$x^{2} = 0$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 5