Calculus Examples

$g (x) = \frac{1}{\sqrt{8 - x}}$

Step 1

Set the radicand in $\sqrt{8 - x}$ greater than or equal to $0$ to find where the expression is defined.

$8 - x \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $8$ from both sides of the inequality.

$- x \geq - 8$

Step 2.2

Divide each term in $- x \geq - 8$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- x \geq - 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 8}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 8}{- 1}$

Step 2.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 8}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 8$ by $- 1$ .

$x \leq 8$

Step 3

Set the denominator in $\frac{1}{\sqrt{8 - x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{8 - x} = 0$

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.

${\sqrt{8 - x}}^{2} = 0^{2}$

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 $\sqrt{8 - x}$ as ${(8 - x)}^{\frac{1}{2}}$ .

${({(8 - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({(8 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({(8 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

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

${(8 - x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

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.

${(8 - x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(8 - x)}^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$8 - x = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$8 - x = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Subtract $8$ from both sides of the equation.

$- x = - 8$

Step 4.3.2

Divide each term in $- x = - 8$ by $- 1$ and simplify.

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Step 4.3.2.1

Divide each term in $- x = - 8$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 8}{- 1}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 8}{- 1}$

Step 4.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 8}{- 1}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Divide $- 8$ by $- 1$ .

$x = 8$

Step 5

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

Interval Notation:

$(- \infty, 8)$

Set-Builder Notation:

${x | x < 8}$

Step 6