Algebra Examples

$y = 2 \tan (x - \frac{π}{4})$

Step 1

Set the argument in $\tan (x - \frac{π}{4})$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$x - \frac{π}{4} = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.1

Add $\frac{π}{4}$ to both sides of the equation.

$x = \frac{π}{2} + π n + \frac{π}{4}$

Step 2.2

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

$x = π n + \frac{π}{2} \cdot \frac{2}{2} + \frac{π}{4}$

Step 2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

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

$x = π n + \frac{π \cdot 2}{2 \cdot 2} + \frac{π}{4}$

Step 2.3.2

Multiply $2$ by $2$ .

$x = π n + \frac{π \cdot 2}{4} + \frac{π}{4}$

Step 2.4

Combine the numerators over the common denominator.

$x = π n + \frac{π \cdot 2 + π}{4}$

Step 2.5

Add $π \cdot 2$ and $π$ .

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Step 2.5.1

Reorder $π$ and $2$ .

$x = π n + \frac{2 \cdot π + π}{4}$

Step 2.5.2

Add $2 \cdot π$ and $π$ .

$x = π n + \frac{3 \cdot π}{4}$

$x = π n + \frac{3 π}{4}$

Step 3

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

Set-Builder Notation:

${x | x \neq π n + \frac{3 π}{4}}$ , 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:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 5

Determine the domain and range.

Domain: ${x | x \neq π n + \frac{3 π}{4}}$ , for any integer $n$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 6