Algebra Examples

$f (x) = {(6 x)}^{\frac{1}{2}} + 3$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt{{(6 x)}^{1}} + 3$

Step 1.2

Anything raised to $1$ is the base itself.

$\sqrt{6 x} + 3$

Step 2

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

$6 x \geq 0$

Step 3

Divide each term in $6 x \geq 0$ by $6$ and simplify.

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Step 3.1

Divide each term in $6 x \geq 0$ by $6$ .

$\frac{6 x}{6} \geq \frac{0}{6}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $6$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{6 x}{6} \geq \frac{0}{6}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{6}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $6$ .

$x \geq 0$

Step 4

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 5

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

Interval Notation:

$[3, \infty)$

Set-Builder Notation:

${y | y \geq 3}$

Step 6

Determine the domain and range.

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

Range: $[3, \infty), {y | y \geq 3}$

Step 7