Determine if Continuous f(x)=8csc(x^2)
The problem asks to ascertain whether the provided function, f(x) = 8*csc(x^2), is continuous. Here, "csc" stands for the cosecant function, which is the reciprocal of the sine function. Continuity, in the context of a function, means that the function does not have any abrupt changes, holes, or jumps in its graph for its domain. To determine if f(x) is continuous, one would generally look at the properties of the cosecant function, potential undefined points, and limits as x approaches any such points within the domain of x^2.
$f \left(\right. x \left.\right) = 8 csc \left(\right. x^{2} \left.\right)$
Answer
Solution:
Determine the Continuity of the Function $f(x) = 8\csc(x^2)$
Step 1: Identify the domain where the function is defined.
Step 1.1: Equate the inside of the cosecant function to $\pi n$ to find where the function is not defined: $x^2 = \pi n$, where $n$ is any integer.
Step 1.2: Isolate $x$ by solving the equation.
Step 1.2.1: Apply the square root to both sides to remove the square on $x$: $x = \pm\sqrt{\pi n}$.
Step 1.2.2: Account for both the positive and negative square roots in the solution.
Step 1.2.2.1: For the positive root, we have: $x = \sqrt{\pi n}$.
Step 1.2.2.2: For the negative root, we have: $x = -\sqrt{\pi n}$.
Step 1.2.2.3: The full set of solutions includes both positive and negative roots: $x = \sqrt{\pi n}, -\sqrt{\pi n}$.
Step 1.3: The domain consists of all $x$ values that keep the function defined. In Set-Builder Notation: $\{ x | x \neq \sqrt{\pi n}, x \neq -\sqrt{\pi n} \}$ for any integer $n$.
Step 2: Because the domain does not include all real numbers, $8\csc(x^2)$ is not continuous for all real numbers. Hence, the function is not continuous.
Knowledge Notes:
To determine the continuity of a function, one must first understand the concept of continuity. A function is continuous at a point if the following three conditions are met:
The function is defined at the point.
The limit of the function as it approaches the point exists.
The limit of the function as it approaches the point is equal to the function's value at that point.
For a function to be continuous over an interval, it must be continuous at every point in that interval.
The cosecant function, $\csc(x)$, is the reciprocal of the sine function, and it is undefined whenever the sine function is zero. Since $\sin(x) = 0$ at $x = n\pi$ for any integer $n$, $\csc(x)$ is undefined at these points.
When dealing with the function $f(x) = 8\csc(x^2)$, we must consider where the inner function $x^2$ causes the cosecant function to be undefined. This occurs when $x^2 = n\pi$, leading to the values of $x$ that are not in the domain of $f(x)$.
Set-Builder Notation is a method of specifying a set of elements that satisfy a certain condition. In this case, it is used to define the domain of $f(x)$ by excluding the points where the function is not defined.
The square root function, $\sqrt{x}$, has two solutions: a positive and a negative root. When solving equations involving squares, both roots must be considered.
In summary, the relevant knowledge points for this problem include understanding the continuity of functions, properties of the cosecant function, Set-Builder Notation, and solving equations involving square roots.
