Calculus Examples

$f (x) = \frac{x}{\sqrt{2 x - 1}}$

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 $x$ .

$0 = \frac{x}{\sqrt{2 x - 1}}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x = 0$

Step 1.2.2

Exclude the solutions that do not make $0 = \frac{x}{\sqrt{2 x - 1}}$ true.

$No solution$

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 $y$ .

$y = \frac{0}{\sqrt{2 (0) - 1}}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{0}{\sqrt{2 (0) - 1}}$

Step 2.2.2

Simplify $\frac{0}{\sqrt{2 (0) - 1}}$ .

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Step 2.2.2.1

Simplify the denominator.

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Step 2.2.2.1.1

Multiply $2$ by $0$ .

$y = \frac{0}{\sqrt{0 - 1}}$

Step 2.2.2.1.2

Subtract $1$ from $0$ .

$y = \frac{0}{\sqrt{- 1}}$

Step 2.2.2.1.3

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

$y = \frac{0}{i}$

Step 2.2.2.2

Multiply the numerator and denominator of $\frac{0}{i}$ by the conjugate of $i$ to make the denominator real.

$y = \frac{0}{i} \cdot \frac{i}{i}$

Step 2.2.2.3

Multiply.

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Step 2.2.2.3.1

Combine.

$y = \frac{0 i}{i i}$

Step 2.2.2.3.2

Multiply $0$ by $i$ .

$y = \frac{0}{i i}$

Step 2.2.2.3.3

Simplify the denominator.

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Step 2.2.2.3.3.1

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

$y = \frac{0}{i^{1} i}$

Step 2.2.2.3.3.2

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

$y = \frac{0}{i^{1} i^{1}}$

Step 2.2.2.3.3.3

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

$y = \frac{0}{i^{1 + 1}}$

Step 2.2.2.3.3.4

Add $1$ and $1$ .

$y = \frac{0}{i^{2}}$

Step 2.2.2.3.3.5

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

$y = \frac{0}{- 1}$

Step 2.2.2.4

Divide $0$ by $- 1$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 0)$

Step 4