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Chapter 2: Problem 76

Use set-builder notation to describe the domain of each of the functions defined. $$ f(x)=\frac{54}{x+65} $$

Short Answer

Expert verified
The domain of \( f(x) \) is \( \{ x \in \mathbb{R} \mid x \neq -65 \} \).

Step by step solution

01

Understand the problem

We need to determine the domain of the function \( f(x) = \frac{54}{x+65} \). The domain of a function is the set of all possible input values (\(x\)) that will produce a valid output.
02

Determine where the function is undefined

The function \( f(x) = \frac{54}{x+65} \) is undefined where the denominator equals zero. This means we need to solve the equation \( x + 65 = 0 \).
03

Solve for the value of x

Solve the equation to find the value of \( x \) that makes the denominator zero: \( x + 65 = 0 \) implies \( x = -65 \). At \( x = -65 \), the function is undefined.
04

Write the domain in set-builder notation

The domain includes all real numbers except \( x = -65 \). In set-builder notation, this is expressed as: \( \{ x \in \mathbb{R} \mid x eq -65 \} \).

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Key Concepts

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

Understanding the Domain of a Function
In mathematics, the domain of a function encompasses all the input values for which the function is defined and outputs a real-number result. For any function, figuring out the domain is crucial because it informs you about the permissible inputs you can use.
To find the domain:
  • Identify the type of function you have (e.g., polynomial, rational).
  • Determine any restrictions that might arise, such as values causing division by zero or negative numbers under a square root.
For a function like \( f(x) = \frac{54}{x+65} \), which is a rational function, the main concern revolves around the denominator. Any value of \( x \) that makes the denominator zero must be excluded from the domain.
Overall, the domain of a rational function is typically all real numbers, except where the denominator is zero.
Exploring Rational Functions
Rational functions are a significant part of algebra and calculus. They are formed by the ratio of two polynomials. More simply, a rational function is any function that can be expressed in the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials.
Unlike polynomials, rational functions can have restrictions on their domains due to denominators:
  • If the polynomial in the denominator, \( Q(x) \), evaluates to zero, the function is undefined at that \( x \) value.
  • The undefined points lead to holes or vertical asymptotes on the graph of the function.
Understanding rational functions requires identifying these restrictions. In our example, \( f(x) = \frac{54}{x+65} \), the rational function must exclude any \( x \) that results in a denominator of zero, ensuring that your function remains valid.
Identifying Undefined Values
Undefined values are crucial to understanding the behavior of rational functions. An undefined value occurs at any \( x \) that makes the function's denominator zero, leading to division by zero鈥攕omething mathematically undefined.
To uncover undefined values:
  • Set the denominator equal to zero and solve for \( x \).
  • Exclude these values from the domain of the function.
For instance, with \( f(x) = \frac{54}{x+65} \):
  • We solve \( x + 65 = 0 \), thus \( x = -65 \) is an undefined value.
  • The domain, therefore, is all \( x \, \in \mathbb{R} \) except where \( x = -65 \).
Leveraging set-builder notation expresses this effectively, helping neatly outline where the function operates correctly: \( \{ x \in \mathbb{R} \mid x eq -65 \} \). Understanding and specifying these undefined values is key to correctly interpreting rational functions.

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