Calculus Examples

$f (x) = \frac{8 x}{9 x - 1}$

Step 1

Set the denominator in $\frac{8 x}{9 x - 1}$ equal to $0$ to find where the expression is undefined.

$9 x - 1 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $1$ to both sides of the equation.

$9 x = 1$

Step 2.2

Divide each term in $9 x = 1$ by $9$ and simplify.

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Step 2.2.1

Divide each term in $9 x = 1$ by $9$ .

$\frac{9 x}{9} = \frac{1}{9}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $9$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{1}{9}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{9}$

Step 3

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

Interval Notation:

$(- \infty, \frac{1}{9}) \cup (\frac{1}{9}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{1}{9}}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \frac{8}{9}) \cup (\frac{8}{9}, \infty)$

Set-Builder Notation:

${y | y \neq \frac{8}{9}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \frac{1}{9}) \cup (\frac{1}{9}, \infty), {x | x \neq \frac{1}{9}}$

Range: $(- \infty, \frac{8}{9}) \cup (\frac{8}{9}, \infty), {y | y \neq \frac{8}{9}}$

Step 6