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Magazine Chapter 1: Problem 2
Find the domain and range of each function. $$f(x)=1-\sqrt{x}$$
Short Answer
Expert verified
Domain: \([0, \infty)\); Range: \((-
fty, 1]\).
Step by step solution
01
Identify Domain Restrictions
The domain of a function involves the set of all possible input values (x-values) for which the function is defined. Since the function involves a square root, we need to identify restrictions: square roots are defined for non-negative numbers. Thus, for \( \sqrt{x} \) to be valid, \( x \geq 0 \).
02
Write the Domain of the Function
Given that the square root requires non-negative inputs, the domain of \( f(x) = 1 - \sqrt{x} \) is all real numbers \( x \) such that \( x \geq 0 \). In interval notation, this is expressed as \([0, \infty)\).
03
Determine Range of the Inner Function
Compute the range of \( \sqrt{x} \) first. For \( x \geq 0 \), \( \sqrt{x} \) outputs non-negative values ranging from 0 to \( \infty\).
04
Transform Inner Range to Find Overall Range
Since \( f(x) = 1 - \sqrt{x} \), we now transform \( \sqrt{x} \) range: \( f(x) = 1 - y \) where \( y \) is any value \( \sqrt{x} \) can take, i.e., from 0 to \( \infty\). Thus, when \( \sqrt{x} = 0 \), \( f(x) = 1 \), and as \( \sqrt{x} \to \infty \), \( f(x) \to -\infty \). Therefore the range of \( f(x) \) is \((-fty, 1]\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Functions
Functions are a foundational concept in mathematics, representing a relationship where each input is associated with exactly one output. In simpler terms, you can think of a function as a machine that takes an input, processes it, and produces an output. It's important to understand a few key aspects of functions:
- Input and Output: The input of the function is commonly referred to as the independent variable, often labeled as \( x \). The output is called the dependent variable, denoted as \( f(x) \) or sometimes \( y \).
- Domain: The domain refers to all possible input values for the function. These are the values for which the function can produce an output.
- Range: The range, on the other hand, describes all possible outputs of the function.
- Notation: Functions are often written in the form \( f(x) \), indicating the function name (\( f \)) and its associated variable (\( x \)).
Square Root Function
The square root function is one of the essential types of functions encountered in algebra. It's written as \( \sqrt{x} \) and represents a number which, when multiplied by itself, gives \( x \). This function has unique characteristics:
- Non-negative Values: The square root function is only defined for non-negative numbers because the square root of a negative number is not a real number. Thus, its domain is \( x \geq 0 \).
- Behavior: For any input \( x \geq 0 \), \( \sqrt{x} \geq 0 \). This is because the result is a non-negative number.
- Graph: The graph of \( \sqrt{x} \) is a gentle upward curve starting from the origin (0, 0) and becomes less steep as \( x \) increases.
Interval Notation
Interval notation is a concise way of expressing a range of values, particularly useful when describing domains and ranges of functions. It is a language of symbols that tells you which numbers are included in a set. Here's how it works:
- Using Brackets: Square brackets \([a, b]\) are used to indicate that the values \( a \) and \( b \) are included in the interval, encompassing every number between and including \( a \) and \( b \). Round brackets \((a, b)\), conversely, indicate that \( a \) and \( b \) are not included, covering every number strictly between them.
- Infinite Intervals: To describe intervals extending to infinity, such as all values greater than a certain number, we use infinity symbols: \( (a, \infty) \) or \( [a, \infty) \). Infinity always uses a round bracket \(()\) since infinity is a concept, not a number we can achieve.
- Complete Domain and Range: For a function like \( f(x) = 1 - \sqrt{x} \), we express the domain as \( [0, \infty) \), meaning all real numbers starting from 0. Its range, on transforming \( \sqrt{x} \) values to \( f(x) \), becomes \((-fty, 1] \), which includes every number less than or equal to 1.
