Find the Domain f(x)=1/( square root of 4x-3)

$f (x) = \frac{1}{\sqrt{4 x - 3}}$

Step 1

Set the radicand in $\sqrt{4 x - 3}$ greater than or equal to $0$ to find where the expression is defined.

$4 x - 3 \geq 0$

Step 2

Solve for $x$ .

Add $3$ to both sides of the inequality.

$4 x \geq 3$

Divide each term in $4 x \geq 3$ by $4$ and simplify.

Divide each term in $4 x \geq 3$ by $4$ .

$\frac{4 x}{4} \geq \frac{3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{3}{4}$

Divide $x$ by $1$ .

$x \geq \frac{3}{4}$

Step 3

Set the denominator in $\frac{1}{\sqrt{4 x - 3}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{4 x - 3} = 0$

Step 4

Solve for $x$ .

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

${\sqrt{4 x - 3}}^{2} = 0^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x - 3}$ as ${(4 x - 3)}^{\frac{1}{2}}$ .

${({(4 x - 3)}^{\frac{1}{2}})}^{2} = 0^{2}$

Simplify the left side.

Simplify ${({(4 x - 3)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 x - 3)}^{\frac{1}{2}})}^{2}$ .

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

${(4 x - 3)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 x - 3)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

${(4 x - 3)}^{1} = 0^{2}$

Simplify.

$4 x - 3 = 0^{2}$

Simplify the right side.

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

$4 x - 3 = 0$

Solve for $x$ .

Add $3$ to both sides of the equation.

$4 x = 3$

Divide each term in $4 x = 3$ by $4$ and simplify.

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 5

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

Interval Notation:

$(\frac{3}{4}, \infty)$

Set-Builder Notation:

${x | x > \frac{3}{4}}$

Step 6

Find the Domain f(x)=1/( square root of 4x-3)

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