Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 77

Chapter 3, Problem 77

Find the domain of each function. g(x) = 4/(x - 7)

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Identify the function given: \(g(x) = \frac{4}{x - 7}\).

Recall that the domain of a function 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 values to exclude: \(x - 7 = 0\).

Solve the equation for \(x\): \(x = 7\).

Conclude that the domain of \(g(x)\) is all real numbers except \(x = 7\), which can be written in interval notation as \((-\infty, 7) \cup (7, \infty)\).

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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 possible input values (x-values) for which the function is defined. It excludes any values that cause undefined expressions, such as division by zero or taking the square root of a negative number in the real number system.

Restrictions from Denominators

When a function includes a fraction, the denominator cannot be zero because division by zero is undefined. To find the domain, identify values of x that make the denominator zero and exclude them from the domain.

Solving Equations to Find Domain Restrictions

To determine domain restrictions, set the denominator equal to zero and solve for x. The solutions are the values that must be excluded from the domain. For g(x) = 4/(x - 7), solve x - 7 = 0 to find the restricted value.

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