Find the Domain and Range x/( square root of x-4)

$\frac{x}{\sqrt{x - 4}}$

Step 1

Set the radicand in $\sqrt{x - 4}$ greater than or equal to $0$ to find where the expression is defined.

$x - 4 \geq 0$

Step 2

Add $4$ to both sides of the inequality.

$x \geq 4$

Step 3

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

$\sqrt{x - 4} = 0$

Step 4

Solve for $x$ .

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x - 4}}^{2} = 0^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 4}$ as ${(x - 4)}^{\frac{1}{2}}$ .

${({(x - 4)}^{\frac{1}{2}})}^{2} = 0^{2}$

Simplify the left side.

Simplify ${({(x - 4)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x - 4)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x - 4)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 4)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

${(x - 4)}^{1} = 0^{2}$

Simplify.

$x - 4 = 0^{2}$

Simplify the right side.

Raising $0$ to any positive power yields $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Step 5

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

Interval Notation:

$(4, \infty)$

Set-Builder Notation:

${x | x > 4}$

Step 6

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

Interval Notation:

$[4, \infty)$

Set-Builder Notation:

${y | y \geq 4}$

Step 7

Determine the domain and range.

Domain: $(4, \infty), {x | x > 4}$

Range: $[4, \infty), {y | y \geq 4}$

Step 8

Find the Domain and Range x/( square root of x-4)

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