Calculus Examples

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

Step 1

Find the domain for $y = \frac{x}{\sqrt{2 x - 1}}$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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Step 1.1

Set the radicand in $\sqrt{2 x - 1}$ greater than or equal to $0$ to find where the expression is defined.

$2 x - 1 \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Add $1$ to both sides of the inequality.

$2 x \geq 1$

Step 1.2.2

Divide each term in $2 x \geq 1$ by $2$ and simplify.

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Step 1.2.2.1

Divide each term in $2 x \geq 1$ by $2$ .

$\frac{2 x}{2} \geq \frac{1}{2}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{1}{2}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{1}{2}$

Step 1.3

Set the denominator in $\frac{x \sqrt{2 x - 1}}{2 x - 1}$ equal to $0$ to find where the expression is undefined.

$2 x - 1 = 0$

Step 1.4

Solve for $x$ .

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Step 1.4.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 1.4.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 1.4.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 1.4.2.2

Simplify the left side.

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Step 1.4.2.2.1

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 1.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 1.5

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

Interval Notation:

$(\frac{1}{2}, \infty)$

Set-Builder Notation:

${x | x > \frac{1}{2}}$

Interval Notation:

$(\frac{1}{2}, \infty)$

Set-Builder Notation:

${x | x > \frac{1}{2}}$

Step 2

To find the radical expression end point, substitute the $x$ value $\frac{1}{2}$ , which is the least value in the domain, into $f (x) = \frac{x \sqrt{2 x - 1}}{2 x - 1}$ .

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Step 2.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

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

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 2.3

Subtract $1$ from $1$ .

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

Step 2.4

The expression contains a division by $0$ . The expression is undefined.

$f (\frac{1}{2}) = Undefined$

Undefined

Step 3

The radical expression end point is $(\frac{1}{2}, Undefined)$ .

$(\frac{1}{2}, Undefined)$

Step 4

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

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Step 4.1

Substitute the $x$ value $1$ into $f (x) = \frac{x \sqrt{2 x - 1}}{2 x - 1}$ . In this case, the point is $(1, 1)$ .

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Step 4.1.1

Replace the variable $x$ with $1$ in the expression.

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $\sqrt{2 (1) - 1}$ by $1$ .

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

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Multiply $2$ by $1$ .

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

Step 4.1.2.2.2

Subtract $1$ from $2$ .

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

Step 4.1.2.3

Simplify the numerator.

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Step 4.1.2.3.1

Multiply $2$ by $1$ .

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

Step 4.1.2.3.2

Subtract $1$ from $2$ .

$f (1) = \frac{\sqrt{1}}{1}$

Step 4.1.2.3.3

Any root of $1$ is $1$ .

$f (1) = \frac{1}{1}$

Step 4.1.2.4

Divide $1$ by $1$ .

$f (1) = 1$

Step 4.1.2.5

The final answer is $1$ .

$y = 1$

Step 4.2

Substitute the $x$ value $2$ into $f (x) = \frac{x \sqrt{2 x - 1}}{2 x - 1}$ . In this case, the point is $(2, \frac{2 \sqrt{3}}{3})$ .

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Step 4.2.1

Replace the variable $x$ with $2$ in the expression.

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

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

Multiply $2$ by $2$ .

$f (2) = \frac{2 \sqrt{4 - 1}}{2 (2) - 1}$

Step 4.2.2.1.2

Subtract $1$ from $4$ .

$f (2) = \frac{2 \sqrt{3}}{2 (2) - 1}$

Step 4.2.2.2

Simplify the denominator.

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Step 4.2.2.2.1

Multiply $2$ by $2$ .

$f (2) = \frac{2 \sqrt{3}}{4 - 1}$

Step 4.2.2.2.2

Subtract $1$ from $4$ .

$f (2) = \frac{2 \sqrt{3}}{3}$

Step 4.2.2.3

The final answer is $\frac{2 \sqrt{3}}{3}$ .

$y = \frac{2 \sqrt{3}}{3}$

Step 4.3

Substitute the $x$ value $3$ into $f (x) = \frac{x \sqrt{2 x - 1}}{2 x - 1}$ . In this case, the point is $(3, \frac{3 \sqrt{5}}{5})$ .

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Step 4.3.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{(3) \sqrt{2 (3) - 1}}{2 (3) - 1}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify the numerator.

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Step 4.3.2.1.1

Multiply $2$ by $3$ .

$f (3) = \frac{3 \sqrt{6 - 1}}{2 (3) - 1}$

Step 4.3.2.1.2

Subtract $1$ from $6$ .

$f (3) = \frac{3 \sqrt{5}}{2 (3) - 1}$

Step 4.3.2.2

Simplify the denominator.

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Step 4.3.2.2.1

Multiply $2$ by $3$ .

$f (3) = \frac{3 \sqrt{5}}{6 - 1}$

Step 4.3.2.2.2

Subtract $1$ from $6$ .

$f (3) = \frac{3 \sqrt{5}}{5}$

Step 4.3.2.3

The final answer is $\frac{3 \sqrt{5}}{5}$ .

$y = \frac{3 \sqrt{5}}{5}$

Step 4.4

The square root can be graphed using the points around the vertex $(0.5, Undefined), (1, 1), (2, 1.15), (3, 1.34)$

$\begin{array}{c} x & y \\ 0.5 & Undefined \\ 1 & 1 \\ 2 & 1.15 \\ 3 & 1.34 \end{array}$

Step 5