Pre-Algebra Examples

$f (x) = \sqrt{e^{x} x}$

Step 1

Find the domain for $y = \sqrt{e^{x} x}$ 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 radicand in $\sqrt{x e^{x}}$ greater than or equal to $0$ to find where the expression is defined.

$x e^{x} \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$e^{x} = 0$

Step 1.2.2

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.3

Set $e^{x}$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $e^{x}$ equal to $0$ .

$e^{x} = 0$

Step 1.2.3.2

Solve $e^{x} = 0$ for $x$ .

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Step 1.2.3.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (0)$

Step 1.2.3.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 1.2.3.2.3

There is no solution for $e^{x} = 0$

No solution

Step 1.2.4

The final solution is all the values that make $x e^{x} \geq 0$ true.

$x = 0$

Step 1.2.5

The solution consists of all of the true intervals.

$x \geq 0$

Step 1.3

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 2

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

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Step 2.1

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

$f (0) = \sqrt{(0) \cdot e^{0}}$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $0$ by $e^{0}$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

$f (0) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 3

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

$(0, 0)$

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

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Step 4.1.1

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

$f (1) = \sqrt{(1) \cdot e^{1}}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $e^{1}$ by $1$ .

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

Step 4.1.2.2

The final answer is $\sqrt{e}$ .

$y = \sqrt{e}$

Step 4.2

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

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Step 4.2.1

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

$f (2) = \sqrt{(2) \cdot e^{2}}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Reorder $2$ and $e^{2}$ .

$f (2) = \sqrt{e^{2} \cdot 2}$

Step 4.2.2.2

Pull terms out from under the radical.

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

Step 4.2.2.3

The final answer is $e \sqrt{2}$ .

$y = e \sqrt{2}$

Step 4.3

The square root can be graphed using the points around the vertex $(0, 0), (1, 1.65), (2, 3.84)$

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 1.65 \\ 2 & 3.84 \end{array}$

Step 5