Find the Domain and Range x=y^6

$x = y^{6}$

Step 1

Rewrite the equation as $y^{6} = x$ .

$y^{6} = x$

Step 2

Take the 6th root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt[6]{x}$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt[6]{x}$

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt[6]{x}$

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt[6]{x}$

$y = - \sqrt[6]{x}$

$y = \sqrt[6]{x}$

$y = - \sqrt[6]{x}$

Step 4

Set the radicand in $\sqrt[6]{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 5

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 6

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

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 7

Determine the domain and range.

Domain: $[0, \infty), {x | x \geq 0}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 8