Find the Domain and Range f(x)=1/(3x^7)

$f (x) = \frac{1}{3 x^{7}}$

Step 1

Set the denominator in $\frac{1}{3 x^{7}}$ equal to $0$ to find where the expression is undefined.

$3 x^{7} = 0$

Step 2

Solve for $x$ .

Divide each term in $3 x^{7} = 0$ by $3$ and simplify.

Divide each term in $3 x^{7} = 0$ by $3$ .

$\frac{3 x^{7}}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x^{7}}{3} = \frac{0}{3}$

Divide $x^{7}$ by $1$ .

$x^{7} = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x^{7} = 0$

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

$x = \sqrt[7]{0}$

Simplify $\sqrt[7]{0}$ .

Rewrite $0$ as $0^{7}$ .

$x = \sqrt[7]{0^{7}}$

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

$x = 0$

Step 3

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 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

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

Set-Builder Notation:

${y | y \neq 0}$

Step 5

Determine the domain and range.

Domain: $(- \infty, 0) \cup (0, \infty), {x | x \neq 0}$

Range: $(- \infty, 0) \cup (0, \infty), {y | y \neq 0}$

Step 6