Find the Domain y=1/( square root of 3x-5)

$y = \frac{1}{\sqrt{3 x - 5}}$

Step 1

Set the radicand in $\sqrt{3 x - 5}$ greater than or equal to $0$ to find where the expression is defined.

$3 x - 5 \geq 0$

Step 2

Solve for $x$ .

Add $5$ to both sides of the inequality.

$3 x \geq 5$

Divide each term in $3 x \geq 5$ by $3$ and simplify.

Divide each term in $3 x \geq 5$ by $3$ .

$\frac{3 x}{3} \geq \frac{5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{5}{3}$

Divide $x$ by $1$ .

$x \geq \frac{5}{3}$

Step 3

Set the denominator in $\frac{1}{\sqrt{3 x - 5}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{3 x - 5} = 0$

Step 4

Solve for $x$ .

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

${\sqrt{3 x - 5}}^{2} = 0^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x - 5}$ as ${(3 x - 5)}^{\frac{1}{2}}$ .

${({(3 x - 5)}^{\frac{1}{2}})}^{2} = 0^{2}$

Simplify the left side.

Simplify ${({(3 x - 5)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 x - 5)}^{\frac{1}{2}})}^{2}$ .

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

${(3 x - 5)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 x - 5)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

${(3 x - 5)}^{1} = 0^{2}$

Simplify.

$3 x - 5 = 0^{2}$

Simplify the right side.

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

$3 x - 5 = 0$

Solve for $x$ .

Add $5$ to both sides of the equation.

$3 x = 5$

Divide each term in $3 x = 5$ by $3$ and simplify.

Divide each term in $3 x = 5$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3}$

Divide $x$ by $1$ .

$x = \frac{5}{3}$

Step 5

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

Interval Notation:

$(\frac{5}{3}, \infty)$

Set-Builder Notation:

${x | x > \frac{5}{3}}$

Step 6

Find the Domain y=1/( square root of 3x-5)

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