Calculus Examples

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

Step 1

Set the argument in $\ln (e^{- x})$ greater than $0$ to find where the expression is defined.

$e^{- x} > 0$

Step 2

Solve for $x$ .

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Step 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 2.2

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

Undefined

Step 2.3

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

No solution

Step 3

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

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

Step 4

Solve for $x$ .

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Step 4.1

Convert the inequality to an equality.

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

Step 4.2

Solve the equation.

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Step 4.2.1

Expand $\ln (e^{- x})$ .

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Step 4.2.1.1

Expand $\ln (e^{- x})$ by moving $- x$ outside the logarithm.

$- x \ln (e)$

Step 4.2.1.2

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

$- x \cdot 1$

Step 4.2.1.3

Multiply $- 1$ by $1$ .

$- x$

Step 4.2.2

The expanded equation is $- x = 0$ .

$- x = 0$

Step 4.2.3

Divide each term in $- x = 0$ by $- 1$ and simplify.

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Step 4.2.3.1

Divide each term in $- x = 0$ by $- 1$ .

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

Step 4.2.3.2

Simplify the left side.

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Step 4.2.3.2.1

Dividing two negative values results in a positive value.

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

Step 4.2.3.2.2

Divide $x$ by $1$ .

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

Step 4.2.3.3

Simplify the right side.

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Step 4.2.3.3.1

Divide $0$ by $- 1$ .

$x = 0$

Step 4.3

The solution consists of all of the true intervals.

$x \leq 0$

Step 5

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

Interval Notation:

$(- \infty, 0]$

Set-Builder Notation:

${x | x \leq 0}$

Step 6

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${y | y \geq 0}$

Step 7

Determine the domain and range.

Domain: $(- \infty, 0], {x | x \leq 0}$

Range: $[0, \infty), {y | y \geq 0}$

Step 8