Problem 18
Question
In Problems 15-24, find the values of \(x \in\) R for which the given functions are both defined and continuous. $$ f(x)=\cos (2 x) $$
Step-by-Step Solution
Verified Answer
The function is both defined and continuous for all real numbers \( x \in \mathbb{R} \).
1Step 1: Determine the Domain of the Function
The function given is \( f(x) = \cos(2x) \). The cosine function \( \cos u \) is defined for all real numbers \( u \). Therefore, \( \cos(2x) \), and consequently \( f(x) \), is defined for all real numbers \( x \). Thus, the domain is all real numbers \( x \in \mathbb{R} \).
2Step 2: Evaluate Continuity of the Function
The cosine function is known to be continuous everywhere on \( \mathbb{R} \). Since \( f(x) = \cos(2x) \) is a composition of the continuous function \( \cos(u) \) and the linear function \( 2x \), \( f(x) \) inherits continuity from the composition. Thus, \( f(x) \) is continuous for all \( x \in \mathbb{R} \).
Key Concepts
Domain of FunctionsTrigonometric FunctionsReal Numbers
Domain of Functions
The domain of a function refers to all the possible input values (x-values) for which the function is defined. In terms of mathematics, functions can behave differently depending on what is inside them. Some functions have restrictions, while others are defined for every real number.
The function provided, \( f(x) = \cos(2x) \), involves the trigonometric function cosine. The cosine function, \( \cos(u) \), can take any real number as input, meaning it's defined everywhere on the set of real numbers, \( \mathbb{R} \). This is primarily because trigonometric functions like cosine are periodic and continuous with a pattern that repeats without restriction on the input.
Hence, the domain of \( f(x) = \cos(2x) \) is simply all real numbers, or expressed in interval notation, \( x \in \mathbb{R} \). There are no values of \( x \) that need to be excluded from the domain as the cosine function does not have discontinuities or undefined points within the real numbers.
The function provided, \( f(x) = \cos(2x) \), involves the trigonometric function cosine. The cosine function, \( \cos(u) \), can take any real number as input, meaning it's defined everywhere on the set of real numbers, \( \mathbb{R} \). This is primarily because trigonometric functions like cosine are periodic and continuous with a pattern that repeats without restriction on the input.
Hence, the domain of \( f(x) = \cos(2x) \) is simply all real numbers, or expressed in interval notation, \( x \in \mathbb{R} \). There are no values of \( x \) that need to be excluded from the domain as the cosine function does not have discontinuities or undefined points within the real numbers.
Trigonometric Functions
Trigonometric functions are fundamental in many areas of mathematics, particularly in geometry and calculus. These functions include sine, cosine, tangent, and their reciprocals—cosecant, secant, and cotangent. They are defined based on right-angled triangles and units in the circle and are periodic, meaning their values repeat at regular intervals.
The cosine function, \( \cos(x) \), is especially notable for its property that it oscillates between -1 and 1. It has a period of \( 2\pi \), meaning that every \( 2\pi \) units along the x-axis, the cosine function repeats its values. This periodic behavior makes trigonometric functions essential in modeling cyclical phenomena like sound waves or seasonal patterns.
For the function \( f(x) = \cos(2x) \), the expression inside the cosine, \( 2x \), alters its frequency. This means the function completes its cycle twice as fast, with a new period of \( \pi \). Despite this change in frequency, the core properties of the cosine function—being defined and continuous over all real numbers—remain intact.
The cosine function, \( \cos(x) \), is especially notable for its property that it oscillates between -1 and 1. It has a period of \( 2\pi \), meaning that every \( 2\pi \) units along the x-axis, the cosine function repeats its values. This periodic behavior makes trigonometric functions essential in modeling cyclical phenomena like sound waves or seasonal patterns.
For the function \( f(x) = \cos(2x) \), the expression inside the cosine, \( 2x \), alters its frequency. This means the function completes its cycle twice as fast, with a new period of \( \pi \). Despite this change in frequency, the core properties of the cosine function—being defined and continuous over all real numbers—remain intact.
Real Numbers
Real numbers encompass both rational numbers (like fractions) and irrational numbers (like \( \pi \) and \( \sqrt{2} \)). They form a continuous set of numbers that occupy the entire number line. This makes them essential for functions that we analyze in calculus and real analysis.
In the context of functions, particularly \( f(x) = \cos(2x) \), real numbers as a domain mean the function can handle any input value along the number line—from negative infinity to positive infinity. There are no breaks or holes in the real numbers, which is why they provide such a comprehensive groundwork for defining function domains.
Understanding that real numbers are a complete, ordered field helps us appreciate why the domain of many functions includes all real numbers, like \( f(x) = \cos(2x) \), and why continuity is easily checked across this unrestricted set. Functions that interact with real numbers seamlessly integrate mathematical concepts that are foundational and broadly applicable across different mathematical disciplines.
In the context of functions, particularly \( f(x) = \cos(2x) \), real numbers as a domain mean the function can handle any input value along the number line—from negative infinity to positive infinity. There are no breaks or holes in the real numbers, which is why they provide such a comprehensive groundwork for defining function domains.
Understanding that real numbers are a complete, ordered field helps us appreciate why the domain of many functions includes all real numbers, like \( f(x) = \cos(2x) \), and why continuity is easily checked across this unrestricted set. Functions that interact with real numbers seamlessly integrate mathematical concepts that are foundational and broadly applicable across different mathematical disciplines.
Other exercises in this chapter
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