Find the domain of each rational expression. See Example 1. (x + 3) / (x - 6)

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Identify the rational expression given, which is \(\frac{x + 3}{x - 6}\).

Recall that the domain of a rational expression includes all real numbers except those that make the denominator zero, because division by zero is undefined.

Set the denominator equal to zero to find the values to exclude: \(x - 6 = 0\).

Solve the equation \(x - 6 = 0\) to find \(x = 6\).

Conclude that the domain is all real numbers except \(x = 6\), which can be written in interval notation as \((-\infty, 6) \cup (6, \infty)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. For rational expressions, the domain excludes values that make the denominator zero, as division by zero is undefined.

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to simplify and analyze these expressions is essential, especially identifying values that cause the denominator to be zero.

Finding Restrictions on the Domain

To find the domain of a rational expression, set the denominator equal to zero and solve for x. The solutions are excluded from the domain because they make the expression undefined. The domain is all real numbers except these excluded values.

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Related Practice

Textbook Question

For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the midpoint M of line segment PQ. See Examples 1 and 2. P(-5, -6), Q(7, -1)

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Textbook Question

Find each sum or difference. See Example 1. -6 + (-13)

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Textbook Question

Work each matching problem.

Match each equation in Column I with a description of its graph from Column II as it relates to the graph of y = x².

I II

a. y = (x - 7)² A. a translation to the left 7 units

b. y = x² - 7 B. a translation to the right 7 units

c. y = 7x² C. a translation up 7 units

d. y = (x + 7)² D. a translation down 7 units