Algebra Examples

$f (x) = \frac{1}{\sqrt{x - 9}}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{1}{\sqrt{x - 9}}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$1 = 0$

Step 1.2.2

Since $1 \neq 0$ , there are no solutions.

No solution

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{1}{\sqrt{(0) - 9}}$

Step 2.2

Simplify $\frac{1}{\sqrt{(0) - 9}}$ .

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Step 2.2.1

Simplify the denominator.

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Step 2.2.1.1

Subtract $9$ from $0$ .

$y = \frac{1}{\sqrt{- 9}}$

Step 2.2.1.2

Rewrite $- 9$ as $- 1 (9)$ .

$y = \frac{1}{\sqrt{- 1 (9)}}$

Step 2.2.1.3

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$y = \frac{1}{\sqrt{- 1} \cdot \sqrt{9}}$

Step 2.2.1.4

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{1}{i \cdot \sqrt{9}}$

Step 2.2.1.5

Rewrite $9$ as $3^{2}$ .

$y = \frac{1}{i \cdot \sqrt{3^{2}}}$

Step 2.2.1.6

Pull terms out from under the radical, assuming positive real numbers.

$y = \frac{1}{i \cdot 3}$

Step 2.2.1.7

Move $3$ to the left of $i$ .

$y = \frac{1}{3 i}$

Step 2.2.2

Multiply the numerator and denominator of $\frac{1}{3 i}$ by the conjugate of $3 i$ to make the denominator real.

$y = \frac{1}{3 i} \cdot \frac{i}{i}$

Step 2.2.3

Multiply.

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Step 2.2.3.1

Combine.

$y = \frac{1 i}{3 i i}$

Step 2.2.3.2

Multiply $i$ by $1$ .

$y = \frac{i}{3 i i}$

Step 2.2.3.3

Simplify the denominator.

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Step 2.2.3.3.1

Add parentheses.

$y = \frac{i}{3 (i i)}$

Step 2.2.3.3.2

Raise $i$ to the power of $1$ .

$y = \frac{i}{3 (i^{1} i)}$

Step 2.2.3.3.3

Raise $i$ to the power of $1$ .

$y = \frac{i}{3 (i^{1} i^{1})}$

Step 2.2.3.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{i}{3 i^{1 + 1}}$

Step 2.2.3.3.5

Add $1$ and $1$ .

$y = \frac{i}{3 i^{2}}$

Step 2.2.3.3.6

Rewrite $i^{2}$ as $- 1$ .

$y = \frac{i}{3 \cdot - 1}$

Step 2.2.4

Multiply $3$ by $- 1$ .

$y = \frac{i}{- 3}$

Step 2.2.5

Move the negative in front of the fraction.

$y = - \frac{i}{3}$

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $None$

Step 4