Trigonometry Examples

$y = \frac{3 x - 1}{x^{2} - 4 x + 4}$

Step 1

Find the domain to determine if the expression is continuous.

Tap for more steps...

Step 1.1

Set the denominator in $\frac{3 x - 1}{x^{2} - 4 x + 4}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 4 x + 4 = 0$

Step 1.2

Solve for $x$ .

Tap for more steps...

Step 1.2.1

Factor using the perfect square rule.

Tap for more steps...

Step 1.2.1.1

Rewrite $4$ as $2^{2}$ .

$x^{2} - 4 x + 2^{2} = 0$

Step 1.2.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 1.2.1.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 2 + 2^{2} = 0$

Step 1.2.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

${(x - 2)}^{2} = 0$

Step 1.2.2

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

$x - 2 = 0$

Step 1.2.3

Add $2$ to both sides of the equation.

$x = 2$

Step 1.3

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 2}$

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 2}$

Step 2

Since the domain is not all real numbers, $\frac{3 x - 1}{x^{2} - 4 x + 4}$ is not continuous over all real numbers.

Not continuous

Step 3