Algebra Examples

f (x) = tan (x - π)

$f (x) = \tan (x - π)$

Step 1

Set the argument in

tan (x - π)

$\tan (x - π)$ equal to

\frac{π}{2} + π n

$\frac{π}{2} + π n$ to find where the expression is undefined.

x - π = \frac{π}{2} + π n

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

n

$n$

Step 2

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 2.1

Add

π

$π$ to both sides of the equation.

x = \frac{π}{2} + π n + π

$x = \frac{π}{2} + π n + π$

Step 2.2

To write

π

$π$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

x = π n + \frac{π}{2} + π \cdot \frac{2}{2}

$x = π n + \frac{π}{2} + π \cdot \frac{2}{2}$

Step 2.3

Combine

π

$π$ and

\frac{2}{2}

$\frac{2}{2}$ .

x = π n + \frac{π}{2} + \frac{π \cdot 2}{2}

$x = π n + \frac{π}{2} + \frac{π \cdot 2}{2}$

Step 2.4

Combine the numerators over the common denominator.

x = π n + \frac{π + π \cdot 2}{2}

$x = π n + \frac{π + π \cdot 2}{2}$

Step 2.5

Add

π

$π$ and

π \cdot 2

$π \cdot 2$ .

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Step 2.5.1

Reorder

π

$π$ and

2

$2$ .

x = π n + \frac{π + 2 \cdot π}{2}

$x = π n + \frac{π + 2 \cdot π}{2}$

Step 2.5.2

Add

π

$π$ and

2 \cdot π

$2 \cdot π$ .

x = π n + \frac{3 π}{2}

$x = π n + \frac{3 π}{2}$

x = π n + \frac{3 π}{2}

$x = π n + \frac{3 π}{2}$

x = π n + \frac{3 π}{2}

$x = π n + \frac{3 π}{2}$

Step 3

The domain is all values of

x

$x$ that make the expression defined.

Set-Builder Notation:

{x ∣ ∣ ∣ x \neq π n + \frac{3 π}{2}}

${x | x \neq π n + \frac{3 π}{2}}$ , for any integer

n

$n$

Step 4

The range is the set of all valid

y

$y$ values. Use the graph to find the range.

Interval Notation:

(- \infty, \infty)

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 5

Determine the domain and range.

Domain:

${x | x \neq π n + \frac{3 π}{2}}$ , for any integer

$n$

Range:

$(- \infty, \infty), {y | y \in ℝ}$

Step 6