Calculus Examples

$f (x) = \frac{e^{x}}{5 + e^{x}}$

Step 1

Set the denominator in $\frac{e^{x}}{5 + e^{x}}$ equal to $0$ to find where the expression is undefined.

$5 + e^{x} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $5$ from both sides of the equation.

$e^{x} = - 5$

Step 2.2

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

$\ln (e^{x}) = \ln (- 5)$

Step 2.3

The equation cannot be solved because $\ln (- 5)$ is undefined.

Undefined

Step 2.4

There is no solution for $e^{x} = - 5$

No solution

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

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

Interval Notation:

$(0, 1)$

Set-Builder Notation:

${y | 0 < y < 1}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: $(0, 1), {y | 0 < y < 1}$

Step 6