Trigonometry Examples

$\frac{\arctan (x^{2} - 4)}{3 x^{2} - 6 x}$

Set the denominator in $\frac{\arctan (x^{2} - 4)}{3 x^{2} - 6 x}$ equal to $0$ to find where the expression is undefined.

$3 x^{2} - 6 x = 0$

Solve for $x$ .

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Factor $3 x$ out of $3 x^{2} - 6 x$ .

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Factor $3 x$ out of $3 x^{2}$ .

$3 x (x) - 6 x = 0$

Factor $3 x$ out of $- 6 x$ .

$3 x (x) + 3 x (- 2) = 0$

Factor $3 x$ out of $3 x (x) + 3 x (- 2)$ .

$3 x (x - 2) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 2 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 2$ equal to $0$ and solve for $x$ .

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Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $3 x (x - 2) = 0$ true.

$x = 0, 2$

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

Interval Notation:

$(- \infty, 0) \cup (0, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 0, 2}$

Since the domain is not all real numbers, $\frac{\arctan (x^{2} - 4)}{3 x^{2} - 6 x}$ is not continuous over all real numbers.

Not continuous