Problem 26
Question
If \(f(x)=\left[\frac{1}{\sqrt{2}}(\cos x+\sin x)\right], 0
Step-by-Step Solution
Verified Answer
There are 4 points of discontinuity.
1Step 1: Analyze the Function Inside Brackets
Consider the expression inside the greatest integer function: \( g(x) = \frac{1}{\sqrt{2}}(\cos x + \sin x) \). To determine the discontinuities, we first analyze where this expression crosses integer values.
2Step 2: Simplify the Expression
The expression \( g(x) \) can be rewritten using the identity \( \cos x + \sin x = \sqrt{2}\sin(x + \frac{\pi}{4}) \). Therefore, \( g(x) = \sin(x + \frac{\pi}{4}) \).
3Step 3: Identify Critical Points of Discontinuity
The greatest integer function \([g(x)]\) is discontinuous at points where \(g(x)\) is an integer. \(g(x) = 0, \pm 1\) within the domain \((0, 2\pi)\). Find \(x\) such that \(\sin(x + \frac{\pi}{4}) = 0, \pm 1\).
4Step 4: Calculate Specific Points
By setting \(\sin(x + \frac{\pi}{4}) = 0\), we get \(x = \frac{3\pi}{4}, \frac{7\pi}{4}\). For \(\sin(x + \frac{\pi}{4}) = 1\), \(x = \frac{\pi}{2}\). For \(\sin(x + \frac{\pi}{4}) = -1\), \(x = \frac{3\pi}{2}\).
5Step 5: Verify Results and Count Discontinuities
Plugging these values into \(f(x) = [g(x)]\), there are no additional points where a change in integer value occurs within \((0, 2\pi)\) other than those calculated. Therefore, there are 4 discontinuities: \(x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\).
Key Concepts
Greatest Integer FunctionTrigonometric IdentitiesFunctions
Greatest Integer Function
The greatest integer function, often symbolized by \( [x] \), is also known as the floor function. This function takes a real number and assigns to it the largest integer that is less than or equal to the number. For example:
For instance, at \( x = 1 \), the function shifts from 0 to 1. This jump causes a discontinuity, and understanding where these jumps occur helps us understand the behavior of functions in problems that include the greatest integer function.
In the given exercise, we see that the function is composed in part by the floor function applied to \( g(x) = \sin(x + \frac{\pi}{4}) \). Analyzing where \( g(x) \) reaches an integer value helps identify points of discontinuity in the overall function.
- \( [3.7] = 3 \)
- \( [5] = 5 \)
- \( [-2.3] = -3 \)
For instance, at \( x = 1 \), the function shifts from 0 to 1. This jump causes a discontinuity, and understanding where these jumps occur helps us understand the behavior of functions in problems that include the greatest integer function.
In the given exercise, we see that the function is composed in part by the floor function applied to \( g(x) = \sin(x + \frac{\pi}{4}) \). Analyzing where \( g(x) \) reaches an integer value helps identify points of discontinuity in the overall function.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. One such useful identity is \( \sin(A + B) = \sin A\cos B + \cos A\sin B \).In solving the exercise above, the equation\( \cos x + \sin x = \sqrt{2}\sin(x + \frac{\pi}{4}) \)is essential.
This transformation is achieved by realizing the form \( \sin(x + A)\). When the coefficients of \( \cos \) and \( \sin \) are equal as in \( \frac{1}{\sqrt{2}}\), the identity allows us to express the sum as a single sine function with a phase shift. Its use simplifies finding specific x-values that lead to integer results, crucial for determining discontinuities in a problem using the greatest integer function. By helping to simplify complex expressions, these identities are vital tools in trigonometry.
This transformation is achieved by realizing the form \( \sin(x + A)\). When the coefficients of \( \cos \) and \( \sin \) are equal as in \( \frac{1}{\sqrt{2}}\), the identity allows us to express the sum as a single sine function with a phase shift. Its use simplifies finding specific x-values that lead to integer results, crucial for determining discontinuities in a problem using the greatest integer function. By helping to simplify complex expressions, these identities are vital tools in trigonometry.
Functions
Functions are mathematical entities that assign unique outputs to given inputs. In simpler terms, for each input into the function, there is a corresponding output. Functions can be thought of as machines where numbers go in, and after processing, numbers come out.
Functions are usually written as \( f(x) \), where \( x \) is the input. The behavior of functions, like where they increase, decrease, or even where they may jump or have breaks like discontinuities, is important to understand. Discontinuities occur where there are abrupt changes in the value of the function; these happen at the points identified in the problem where the expression \( \sin(x + \frac{\pi}{4}) \) equals integer results within the greatest integer function.
Understanding how functions behave, transform, and interact with operations such as the greatest integer function in particular problems helps us predict and solve for these unique points.
Functions are usually written as \( f(x) \), where \( x \) is the input. The behavior of functions, like where they increase, decrease, or even where they may jump or have breaks like discontinuities, is important to understand. Discontinuities occur where there are abrupt changes in the value of the function; these happen at the points identified in the problem where the expression \( \sin(x + \frac{\pi}{4}) \) equals integer results within the greatest integer function.
Understanding how functions behave, transform, and interact with operations such as the greatest integer function in particular problems helps us predict and solve for these unique points.
Other exercises in this chapter
Problem 24
If \(f(x)=\int_{0}^{x} t \cos \frac{1}{t} d t\), then the number of points of discontinuity of \(f(x)\) in the interval \((0, \pi)\) is (A) 1 (B) 2 (C) 0 (D) No
View solution Problem 25
If \(f(x)=(-1)^{\left[x^{2}\right]}\), where \([.]\) denotes the greatest integer function, then (A) \(f(x)\) is discontinuous for \(x=n^{1 / 3}\), where \(n \i
View solution Problem 27
Let \(f(x)=a[x]+b e^{|x|}+c|x|^{2}\), where \(a, b\) and \(c\) are real constants. If \(f(x)\) is differentiable at \(x=0\), then (A) \(b=0, c=0, a \in R\) (B)
View solution Problem 28
If \(f(x)=[x] \sin \left(\frac{\pi}{[x+1]}\right)\), where \([.]\) denotes the greatest integer function, then the points of discontinuity of \(f\) in the domai
View solution