Calculus Examples

$f (x) = \frac{1}{4 x - 1}$ , $[- 2, 10]$

Step 1

To find whether the function is continuous on $[- 2, 10]$ or not, find the domain of $f (x) = \frac{1}{4 x - 1}$ .

Tap for more steps...

Step 1.1

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

$4 x - 1 = 0$

Step 1.2

Solve for $x$ .

Tap for more steps...

Step 1.2.1

Add $1$ to both sides of the equation.

$4 x = 1$

Step 1.2.2

Divide each term in $4 x = 1$ by $4$ and simplify.

Tap for more steps...

Step 1.2.2.1

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Step 1.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 1.3

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

Interval Notation:

$(- \infty, \frac{1}{4}) \cup (\frac{1}{4}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{1}{4}}$

Interval Notation:

$(- \infty, \frac{1}{4}) \cup (\frac{1}{4}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{1}{4}}$

Step 2

$f (x)$ is not continuous on $[- 2, 10]$ because $\frac{1}{4}$ is not in the domain of $f (x) = \frac{1}{4 x - 1}$ .

The function is not continuous.

Step 3