Calculus Examples

$f (x) = \frac{x + 3}{4 x - 5}$

Step 1

Set the denominator in $\frac{x + 3}{4 x - 5}$ equal to $0$ to find where the expression is undefined.

$4 x - 5 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $5$ to both sides of the equation.

$4 x = 5$

Step 2.2

Divide each term in $4 x = 5$ by $4$ and simplify.

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Step 2.2.1

Divide each term in $4 x = 5$ by $4$ .

$\frac{4 x}{4} = \frac{5}{4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{5}{4}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{4}$

Step 3

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

Interval Notation:

$(- \infty, \frac{5}{4}) \cup (\frac{5}{4}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{5}{4}}$

Step 4

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

Interval Notation:

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

Set-Builder Notation:

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

Step 5

Determine the domain and range.

Domain: $(- \infty, \frac{5}{4}) \cup (\frac{5}{4}, \infty), {x | x \neq \frac{5}{4}}$

Range: $(- \infty, \frac{1}{4}) \cup (\frac{1}{4}, \infty), {y | y \neq \frac{1}{4}}$

Step 6