Precalculus Examples

$f (x) = 2 \sec (x) - 3$

Step 1

Set $2 \sec (x) - 3$ equal to $0$ .

$2 \sec (x) - 3 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $3$ to both sides of the equation.

$2 \sec (x) = 3$

Step 2.2

Divide each term in $2 \sec (x) = 3$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 \sec (x) = 3$ by $2$ .

$\frac{2 \sec (x)}{2} = \frac{3}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 \sec (x)}{2} = \frac{3}{2}$

Step 2.2.2.1.2

Divide $\sec (x)$ by $1$ .

$\sec (x) = \frac{3}{2}$

Step 2.3

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

$x = arcsec (\frac{3}{2})$

Step 2.4

Simplify the right side.

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Step 2.4.1

Evaluate $arcsec (\frac{3}{2})$ .

$x = 0.84106867$

Step 2.5

The secant 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.

$x = 2 (3.14159265) - 0.84106867$

Step 2.6

Solve for $x$ .

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Step 2.6.1

Remove parentheses.

$x = 2 (3.14159265) - 0.84106867$

Step 2.6.2

Simplify $2 (3.14159265) - 0.84106867$ .

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Step 2.6.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 0.84106867$

Step 2.6.2.2

Subtract $0.84106867$ from $6.2831853$ .

$x = 5.44211663$

Step 2.7

Find the period of $\sec (x)$ .

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Step 2.7.1

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

$\frac{2 π}{| b |}$

Step 2.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.8

The period of the $\sec (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 0.84106867 + 2 π n, 5.44211663 + 2 π n$ , for any integer $n$

Step 3