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Chapter 12: Problem 79

Find the domain of each function. $$f(x)=\frac{x-3}{x+7}$$

Short Answer

Expert verified
The domain is \(x eq -7\) or \((-\infty, -7) \cup (-7, \infty)\).

Step by step solution

01

- Identify the Type of Function

Recognize that this is a rational function of the form \(f(x)= \frac{P(x)}{Q(x)} \) where \(P(x) = x - 3\) and \(Q(x) = x + 7\).
02

- Identify Restrictions on the Denominator

The domain of any rational function excludes values that make the denominator zero. Set the denominator equal to zero and solve for \(x\): \(Q(x) = x + 7 = 0\).
03

- Solve for the Restricted Value

Solve the equation \(x + 7 = 0\) to find the restricted value: \(x = -7\).
04

- State the Domain

The domain of the function includes all real numbers except for the value that makes the denominator zero. Therefore, the domain is \(x eq -7\). In interval notation, the domain is \((-\infty, -7) \cup (-7, \infty)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Function
A rational function is a type of function represented as a fraction with a polynomial in both the numerator and the denominator. Mathematically, it takes the form \(f(x)= \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. For example, in the given function \(f(x) = \frac{x-3}{x+7}\), \(P(x) = x - 3\) and \(Q(x) = x + 7\). The behavior of rational functions can vary greatly due to their polynomial components, leading to different restrictions and characteristics.
Domain Restrictions
An essential aspect of working with rational functions is identifying the domain. The domain includes all real numbers for which the function is defined and valid. For rational functions, the primary restriction occurs when the denominator (\(Q(x)\)) equals zero, as dividing by zero is undefined in mathematics.
To find the restriction, set the denominator equal to zero and solve for \(x\). For example, with \(f(x)= \frac{x-3}{x+7}\), solving \(x + 7 = 0\) gives \(x = -7\). This means the function is undefined at \(x = -7\), so it must be excluded from the domain.
The domain is then all real numbers except \(x = -7\).
Interval Notation
Interval notation is a concise way to represent a set of numbers along a number line. It uses parentheses or brackets to indicate the endpoints of the intervals.
  • Parentheses \(()\) are used to denote that an endpoint is not included in the interval. For instance, \((-7, \infty)\) means all numbers greater than -7, but not -7 itself.
  • Brackets \([]\) mean the endpoint is included. For instance, \([-7, \infty)\) includes -7 and all numbers greater than -7.
When denoting the domain of the given rational function \(f(x) = \frac{x-3}{x+7}\), we exclude the value -7. Therefore, the domain in interval notation is written as \((-\infty, -7) \cup (-7, \infty)\), signifying all real numbers except -7.

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Most popular questions from this chapter

Classify each of the following as true or false. Assume a, \(x, P,\) and \(Q>0, a \neq 1\). $$\log _{a}\left(Q+Q^{2}\right)=\log _{a} Q+\log _{a}(Q+1)$$

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Solve for \(x\) $$ \log (492 x)=5.728 $$

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Solve. $$ \log _{3} x=2 $$
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