Trigonometry Examples

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

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

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

$e^{x^{2}} \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

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

$\ln (e^{x^{2}}) \geq \ln (0)$

Step 1.2.2

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

Undefined

Step 1.2.3

There is no solution for $e^{x^{2}} \geq 0$

No solution

Step 1.3

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

$\sqrt{e^{x^{2}}} = 0$

Step 1.4

Solve for $x$ .

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Step 1.4.1

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

${\sqrt{e^{x^{2}}}}^{2} = 0^{2}$

Step 1.4.2

Simplify each side of the equation.

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Step 1.4.2.1

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

${(e^{\frac{x^{2}}{2}})}^{2} = 0^{2}$

Step 1.4.2.2

Simplify the left side.

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Step 1.4.2.2.1

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

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Step 1.4.2.2.1.1

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

$e^{\frac{x^{2}}{2} \cdot 2} = 0^{2}$

Step 1.4.2.2.1.2

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.2.1

Cancel the common factor.

$e^{\frac{x^{2}}{2} \cdot 2} = 0^{2}$

Step 1.4.2.2.1.2.2

Rewrite the expression.

$e^{x^{2}} = 0^{2}$

Step 1.4.2.3

Simplify the right side.

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Step 1.4.2.3.1

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

$e^{x^{2}} = 0$

Step 1.4.3

Solve for $x$ .

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Step 1.4.3.1

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

$\ln (e^{x^{2}}) = \ln (0)$

Step 1.4.3.2

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

Undefined

Step 1.4.3.3

There is no solution for $e^{x^{2}} = 0$

No solution

Step 1.5

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2

Since the domain is all real numbers, $\frac{1}{\sqrt{e^{x^{2}}}} - 1$ is continuous over all real numbers.

Continuous

Step 3