Precalculus Examples

$f (x) = \frac{1}{\sqrt{5 x + 7}}$

Step 1

Set the radicand in $\sqrt{5 x + 7}$ greater than or equal to $0$ to find where the expression is defined.

$5 x + 7 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $7$ from both sides of the inequality.

$5 x \geq - 7$

Step 2.2

Divide each term in $5 x \geq - 7$ by $5$ and simplify.

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Step 2.2.1

Divide each term in $5 x \geq - 7$ by $5$ .

$\frac{5 x}{5} \geq \frac{- 7}{5}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $5$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} \geq \frac{- 7}{5}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 7}{5}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{7}{5}$

Step 3

Set the denominator in $\frac{1}{\sqrt{5 x + 7}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{5 x + 7} = 0$

Step 4

Solve for $x$ .

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Step 4.1

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

${\sqrt{5 x + 7}}^{2} = 0^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x + 7}$ as ${(5 x + 7)}^{\frac{1}{2}}$ .

${({(5 x + 7)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({(5 x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({(5 x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

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

${(5 x + 7)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${(5 x + 7)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(5 x + 7)}^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$5 x + 7 = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$5 x + 7 = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Subtract $7$ from both sides of the equation.

$5 x = - 7$

Step 4.3.2

Divide each term in $5 x = - 7$ by $5$ and simplify.

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Step 4.3.2.1

Divide each term in $5 x = - 7$ by $5$ .

$\frac{5 x}{5} = \frac{- 7}{5}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $5$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 7}{5}$

Step 4.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{5}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Move the negative in front of the fraction.

$x = - \frac{7}{5}$

Step 5

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

Interval Notation:

$(- \frac{7}{5}, \infty)$

Set-Builder Notation:

${x | x > - \frac{7}{5}}$

Step 6