Trigonometry Examples

$f (x) = \frac{5}{3} \cdot \frac{4}{5^{x}}$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the denominator in $\frac{4}{5^{x}}$ equal to $0$ to find where the expression is undefined.

$5^{x} = 0$

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.

$\ln (5^{x}) = \ln (0)$

Step 1.2.2

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

Undefined

Step 1.2.3

There is no solution for $5^{x} = 0$

No solution

No solution

Step 1.3

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, $\frac{5}{3} \cdot \frac{4}{5^{x}}$ is continuous over all real numbers.

Continuous

Step 3

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$