Find the Domain and Range f(x)=3/(cos(4x))

$f (x) = \frac{3}{\cos (4 x)}$

Step 1

Set the denominator in $\frac{3}{\cos (4 x)}$ equal to $0$ to find where the expression is undefined.

$\cos (4 x) = 0$

Step 2

Solve for $x$ .

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$4 x = \arccos (0)$

Simplify the right side.

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$4 x = \frac{π}{2}$

Divide each term in $4 x = \frac{π}{2}$ by $4$ and simplify.

Divide each term in $4 x = \frac{π}{2}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{π}{2}}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{π}{2}}{4}$

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{4}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{4}$

Multiply $\frac{π}{2} \cdot \frac{1}{4}$ .

Multiply $\frac{π}{2}$ by $\frac{1}{4}$ .

$x = \frac{π}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{π}{8}$

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$4 x = 2 π - \frac{π}{2}$

Solve for $x$ .

Simplify.

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4 x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Combine $2 π$ and $\frac{2}{2}$ .

$4 x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Combine the numerators over the common denominator.

$4 x = \frac{2 π \cdot 2 - π}{2}$

Multiply $2$ by $2$ .

$4 x = \frac{4 π - π}{2}$

Subtract $π$ from $4 π$ .

$4 x = \frac{3 π}{2}$

Divide each term in $4 x = \frac{3 π}{2}$ by $4$ and simplify.

Divide each term in $4 x = \frac{3 π}{2}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{3 π}{2}}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{3 π}{2}}{4}$

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{4}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{4}$

Multiply $\frac{3 π}{2} \cdot \frac{1}{4}$ .

Multiply $\frac{3 π}{2}$ by $\frac{1}{4}$ .

$x = \frac{3 π}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{3 π}{8}$

Find the period of $\cos (4 x)$ .

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$\frac{π}{2}$

The period of the $\cos (4 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = \frac{π}{8} + \frac{π n}{2}, \frac{3 π}{8} + \frac{π n}{2}$ , for any integer $n$

Consolidate the answers.

$x = \frac{π}{8} + \frac{π n}{4}$ , for any integer $n$

Step 3

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

Set-Builder Notation:

${x | x \neq \frac{π}{8} + \frac{π n}{4}}$ , for any integer $n$

Step 4

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

Interval Notation:

$(- \infty, - 1] \cup [1, \infty)$

Set-Builder Notation:

${y | y \leq - 1, y \geq 1}$

Step 5

Determine the domain and range.

Domain: ${x | x \neq \frac{π}{8} + \frac{π n}{4}}$ , for any integer $n$

Range: $(- \infty, - 1] \cup [1, \infty), {y | y \leq - 1, y \geq 1}$

Step 6

Find the Domain and Range f(x)=3/(cos(4x))

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