Precalculus Examples

$f (x) = \frac{x^{2} + x - 2}{x^{2} - 3 x - 4}$

Step 1

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

$x^{2} - 3 x - 4 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $x^{2} - 3 x - 4$ using the AC method.

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Step 2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 2.1.2

Write the factored form using these integers.

$(x - 4) (x + 1) = 0$

Step 2.2

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

$x - 4 = 0$

$x + 1 = 0$

Step 2.3

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.5

The final solution is all the values that make $(x - 4) (x + 1) = 0$ true.

$x = 4, - 1$

Step 3

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

Interval Notation:

$(- \infty, - 1) \cup (- 1, 4) \cup (4, \infty)$

Set-Builder Notation:

${x | x \neq - 1, 4}$

Step 4

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

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 5

Determine the domain and range.

Domain: $(- \infty, - 1) \cup (- 1, 4) \cup (4, \infty), {x | x \neq - 1, 4}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 6