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Chapter 1: Problem 8

Find the discontinuities, if any. $$f(x)=\sqrt{2+\tan ^{2} x}$$

Short Answer

Expert verified
Discontinuities are at \( x = \frac{\pi}{2} + k\pi \), integer \( k \).

Step by step solution

01

Understand the Function

The function given is \( f(x) = \sqrt{2 + \tan^2 x} \). To identify discontinuities, we need to determine where the square root and tangent functions cause issues, such as undefined values or non-real numbers.
02

Square Root Function Analysis

The expression under the square root, \( 2+\tan^2 x \), must be non-negative for \( f(x) \) to be defined. Since \( \tan^2 x \geq 0 \) for all \( x \), \( 2+\tan^2 x \geq 2 \) is always non-negative. Therefore, the square root itself does not cause any discontinuities.
03

Analyze the Tangent Function

The tangent function, \( \tan x \), is undefined where \( \cos x = 0 \). This occurs at odd multiples of \( \frac{\pi}{2} \) (i.e., \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer). At these points, \( \tan x \) is undefined, leading to undefined values for \( f(x) \) as well.
04

Conclusion: Identify Discontinuities

The only discontinuities occur where the tangent function is undefined, due to division by zero in \( \tan x = \frac{\sin x}{\cos x} \). Thus, \( f(x) \) is discontinuous at odd multiples of \( \frac{\pi}{2} \): \( x = \frac{\pi}{2} + k\pi \) for integer \( k \).

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Key Concepts

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

Tangent Function
The tangent function, denoted as \( \tan x \), is a fundamental trigonometric function. It is mathematically represented as the ratio of the sine function \( \sin x \) to the cosine function \( \cos x \). Due to this ratio form, \( \tan x \) becomes undefined whenever \( \cos x \) equals zero. This situation occurs at specific values of \( x \), namely when \( x = \frac{\pi}{2} + k\pi \), where \( k \) is any integer. At these points, the cosine function is zero, leading to division by zero, which results in the tangent becoming undefined.

  • The periodicity of the tangent function is \( \pi \), meaning its pattern repeats every \( \pi \) units.
  • It exhibits vertical asymptotes at each undefined point.
Understanding these properties of the tangent function helps in analyzing its impact on the overall function \( f(x) = \sqrt{2 + \tan^2 x} \).
Square Root Function
The square root function is symbolized by \( \sqrt{} \), which operates by extracting the non-negative root of a given expression. When dealing with square roots, it is crucial for the expression inside, called the radicand, to be non-negative. This means any negative values under a square root would result in non-real numbers. For the function \( f(x) = \sqrt{2 + \tan^2 x} \), the radicand is \( 2 + \tan^2 x \).

  • Because \( \tan^2 x \) is always non-negative, \( 2 + \tan^2 x \) is always greater than or equal to 2. Therefore, it does not cause any discontinuities.
  • The domain of the square root function here is all real numbers where the radicand remains non-negative.
Thus, in this context, the square root imposes no additional constraints beyond those of the tangent function.
Undefined Values
Undefined values in a function typically arise from division by zero or taking the square root of negative numbers. For the function \( f(x) = \sqrt{2 + \tan^2 x} \), undefined values originate due to the behavior of the tangent function. As previously mentioned, \( \tan x \) is undefined wherever its denominator, \( \cos x \), hits zero, which are the points \( x = \frac{\pi}{2} + k\pi \).

  • At these points, \( \tan x \) would lead to an undefined situation, thereby affecting \( f(x) \).
  • These critical points are also where the square root function encounters discontinuities in this context.
  • Discontinuities often imply gaps or breaks in the function's graph, precisely where these undefined values occur.
Grasping how undefined values affect functions allows a deeper understanding of where and why discontinuities happen in mathematical expressions.

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