Calculus Examples

$f (x) = (\frac{3 e^{x}}{e^{x} - 19})$

Step 1

Remove parentheses.

$\frac{3 e^{x}}{e^{x} - 19}$

Step 2

Set the denominator in $\frac{3 e^{x}}{e^{x} - 19}$ equal to $0$ to find where the expression is undefined.

$e^{x} - 19 = 0$

Step 3

Solve for $x$ .

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Step 3.1

Add $19$ to both sides of the equation.

$e^{x} = 19$

Step 3.2

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

$\ln (e^{x}) = \ln (19)$

Step 3.3

Expand the left side.

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Step 3.3.1

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

$x \ln (e) = \ln (19)$

Step 3.3.2

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

$x \cdot 1 = \ln (19)$

Step 3.3.3

Multiply $x$ by $1$ .

$x = \ln (19)$

Step 4

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

Interval Notation:

$(- \infty, \ln (19)) \cup (\ln (19), \infty)$

Set-Builder Notation:

${x | x \neq \ln (19)}$

Step 5

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

Interval Notation:

$(- \infty, 0) \cup (3, \infty)$

Set-Builder Notation:

${y | y < 0, y > 3}$

Step 6

Determine the domain and range.

Domain: $(- \infty, \ln (19)) \cup (\ln (19), \infty), {x | x \neq \ln (19)}$

Range: $(- \infty, 0) \cup (3, \infty), {y | y < 0, y > 3}$

Step 7