Calculus Examples

$\ln (\sqrt{x})$

Step 1

Find the domain for $y = \ln (\sqrt{x})$ 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 argument in $\ln (\sqrt{x})$ greater than $0$ to find where the expression is defined.

$\sqrt{x} > 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

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

${\sqrt{x}}^{2} > 0^{2}$

Step 1.2.2

Simplify each side of the inequality.

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Step 1.2.2.1

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

${(x^{\frac{1}{2}})}^{2} > 0^{2}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

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

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Step 1.2.2.2.1.1

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

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Step 1.2.2.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} > 0^{2}$

Step 1.2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} > 0^{2}$

Step 1.2.2.2.1.1.2.2

Rewrite the expression.

$x^{1} > 0^{2}$

Step 1.2.2.2.1.2

Simplify.

$x > 0^{2}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

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

$x > 0$

Step 1.2.3

Find the domain of $\sqrt{x}$ .

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Step 1.2.3.1

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

$x \geq 0$

Step 1.2.3.2

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

$[0, \infty)$

Step 1.2.4

The solution consists of all of the true intervals.

$x > 0$

Step 1.3

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

$x \geq 0$

Step 1.4

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Step 2

To find the radical expression end point, substitute the $x$ value $0$ , which is the least value in the domain, into $f (x) = \ln (\sqrt{x})$ .

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Step 2.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \ln (\sqrt{0})$

Step 2.2

Remove parentheses.

$f (0) = \ln (\sqrt{0})$

Step 2.3

Rewrite $0$ as $0^{2}$ .

$f (0) = \ln (\sqrt{0^{2}})$

Step 2.4

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

$f (0) = \ln (0)$

Step 2.5

The natural logarithm of zero is undefined.

$f (0) = Undefined$

Undefined

Step 3

The radical expression end point is $(0, Undefined)$ .

$(0, Undefined)$

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 $f (x) = \ln (\sqrt{x})$ . In this case, the point is $(1, 0)$ .

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Step 4.1.1

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

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Remove parentheses.

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

Step 4.1.2.2

Any root of $1$ is $1$ .

$f (1) = \ln (1)$

Step 4.1.2.3

The natural logarithm of $1$ is $0$ .

$f (1) = 0$

Step 4.1.2.4

The final answer is $0$ .

$y = 0$

Step 4.2

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

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Step 4.2.1

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

$f (2) = \ln (\sqrt{2})$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Remove parentheses.

$f (2) = \ln (\sqrt{2})$

Step 4.2.2.2

The final answer is $\ln (\sqrt{2})$ .

$y = \ln (\sqrt{2})$

Step 4.3

The square root can be graphed using the points around the vertex $(0, Undefined), (1, 0), (2, 0.35)$

$\begin{array}{c} x & y \\ 0 & Undefined \\ 1 & 0 \\ 2 & 0.35 \end{array}$

Step 5