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Chapter 2: Problem 39

Find the domain of the function. $$f(x)=2 x, \quad-1 \leq x \leq 5$$

Short Answer

Expert verified
The domain of the function is [-1, 5].

Step by step solution

01

Understanding the Function

The function given is a linear function, specifically, \( f(x) = 2x \). Since it's a linear function, it doesn't have any restrictions that come from operations like division by zero or square roots of negative numbers.
02

Analyzing the Domain Constraints

The exercise provides an explicit domain with constraints: \(-1 \leq x \leq 5\). This means only values of \(x\) from \(-1\) to \(5\), inclusive of both endpoints, are allowed for this function.
03

Writing the Domain

Based on the constraints provided, the domain of \( f(x) \) is represented as an interval. The domain is \([-1, 5]\), which includes all real numbers \(x\) from \(-1\) to \(5\), including \(-1\) and \(5\) themselves.

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Key Concepts

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

Linear Function
A linear function is one of the simplest and most commonly encountered types of functions. When we talk about a linear function, we're referring to a function that can be written in the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants. In our specific example, the function given is \( f(x) = 2x \), which can be seen as having a slope of 2 and a y-intercept of 0 since there is no additional constant term added. Linear functions have some key characteristics:
  • They are represented by a straight line when graphed on a coordinate plane.
  • Every increase in \( x \) results in a consistent increase in \( f(x) \), determined by the slope.
  • They do not include any operations like division by zero, which means they are continuous everywhere they are defined.
The simplicity of linear functions makes them easy to work with, especially when determining domains and ranges.
Interval Notation
Interval notation is a mathematical notation used to specify a range of values. It is particularly useful when describing the domain of a function, as it provides a clear and concise way to express the set of all possible input values (or outputs, in the case of a range). In interval notation, the domain for the function provided in our exercise is written as \([-1, 5]\). This notation tells us several things:
  • The brackets \([ \) and \( ] \) are used to denote that the endpoints are included, which is what we call a "closed interval."
  • If we were not including an endpoint, we would use parentheses, e.g., \((-1, 5] \) would mean all values from just above -1 to 5, excluding -1 itself.
  • Both real numbers and infinity can be represented using interval notation, though infinity is never closed, represented with parentheses like \(( -\infty, \infty )\).
This way of expressing domains is compact and avoids the need to list out each potential value, making it easier and neater, especially when dealing with large or infinite intervals.
Domain Constraints
Domain constraints are limitations on the input values \( x \) of a function. These constraints define where the function is allowed to operate. They are essential because they ensure the function is properly defined and can produce real and meaningful outputs. For any given function, determining domain constraints can be critical.
  • Some functions, like linear ones, generally have less strict domain constraints as they are defined for all real numbers.
  • However, specific problems may impose constraints based on context, like the range given \(-1 \leq x \leq 5\) in our exercise.
  • Constraints may arise from practical scenarios, such as defining the domain only for positive numbers or a particular finite range of interest.
Recognizing and applying domain constraints is a foundational skill in mathematics, ensuring accurate modeling and interpretation of real-world phenomena.

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