Calculus Examples

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

Step 1

Set the argument in $\arcsin (e^{x})$ greater than or equal to $- 1$ to find where the expression is defined.

$e^{x} \geq - 1$

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.

$\ln (e^{x}) \geq \ln (- 1)$

Step 2.2

The equation cannot be solved because $\ln (- 1)$ is undefined.

Undefined

Step 2.3

There is no solution for $e^{x} \geq - 1$

No solution

Step 3

Set the argument in $\arcsin (e^{x})$ less than or equal to $1$ to find where the expression is defined.

$e^{x} \leq 1$

Step 4

Solve for $x$ .

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Step 4.1

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

$\ln (e^{x}) \leq \ln (1)$

Step 4.2

Expand the left side.

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Step 4.2.1

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

$x \ln (e) \leq \ln (1)$

Step 4.2.2

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

$x \cdot 1 \leq \ln (1)$

Step 4.2.3

Multiply $x$ by $1$ .

$x \leq \ln (1)$

Step 4.3

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

$x \leq 0$

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