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Chapter 2: Problem 14

At what points are the functions continuous? $$ y=\frac{1}{(x+2)^{2}}+4 $$

Short Answer

Expert verified
The function is continuous for all \( x \neq -2 \).

Step by step solution

01

Understanding the Problem

The given function is \( y = \frac{1}{(x+2)^2} + 4 \). To determine the points at which this function is continuous, we need to find any points where the function is not defined or could have discontinuities.
02

Identify Points of Discontinuity

The function is a rational function. It appears to have a potential problem when \( x+2 = 0 \), since division by zero is undefined. Thus, we set \( x+2 = 0 \) which gives \( x = -2 \), indicating a point where the function might be discontinuous.
03

Determine Continuity at Other Points

A rational function is continuous at all points where it is defined. Since the issue was only with \( x = -2 \), the function is continuous for all \( x eq -2 \). Apart from this point, all other points are continuous.

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Key Concepts

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

Rational Functions
Rational functions are mathematical expressions involving the division of two polynomials. They have the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomial functions. Since these functions involve division, it鈥檚 important to identify the denominator. This is because a rational function is undefined when the denominator equals zero, leading to potential discontinuities. In our original exercise, the function \( y = \frac{1}{(x+2)^2} + 4 \) is rational. Here, the denominator is \((x+2)^2\), and understanding its behavior is key to determining continuity.
Discontinuities
Discontinuities occur in functions where there are abrupt changes in value. For rational functions, these happen where the denominator equals zero, as was mentioned earlier, because you cannot divide by zero. In our specific example, the discontinuity is at \( x = -2 \), found by setting \( x+2 = 0 \). These points are telling us where the function 'breaks' and cannot be evaluated as usual, causing a gap in the graph. Discontinuities can signal a point where the function needs special attention.
Continuity Definition
A function is continuous if you can draw its graph without lifting your pencil. This means the function goes smoothly from point to point without breaks or jumps. Mathematically, a function \( f(x) \) is continuous at a point \( c \) if the following conditions are met:
  • \( f(c) \) is defined.
  • \( \lim_{x \to c} f(x) \, \text{exists} \).
  • \( \lim_{x \to c} f(x) = f(c) \).
In the context of our exercise, except for \( x = -2 \), the function is continuous because all other points meet these continuity criteria.
Undefined Points
Undefined points in a function occur when it includes operations that have no mathematical meaning, such as division by zero. These points are crucial because they indicate exact places where continuity is disrupted. For rational functions like \( y = \frac{1}{(x+2)^2} + 4 \), you discover undefined points by setting the denominator equal to zero鈥攆or our specific problem, \( x+2 = 0 \) leads to \( x = -2 \). At this value, the rational part of the function cannot be calculated, so the function has no value, indicating a discontinuity at this point.

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Most popular questions from this chapter

One-sided limits \begin{equation}f(x)=\left\\{\begin{array}{ll}{x^{2} \sin (1 / x),} & {x<0} \\\ {\sqrt{x},} & {x>0}.\end{array}\right.\end{equation} Find (a) \(\lim _{x \rightarrow 0^{+}} f(x)\) and (b) \(\lim _{x \rightarrow 0 ^{-}} f(x) ;\) then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about \(\lim _{x \rightarrow 0} f(x) ?\) Give reasons for your answer.

Suppose that \(f\) is an even function of \(x .\) Does knowing that \(\lim _{x \rightarrow 2^{-}} f(x)=7\) tell you anything about either \(\lim _{x \rightarrow-2^{-}} f(x)\) or \(\lim _{x \rightarrow-2^{+}} f(x) ?\) Give reasons for your answer.

Suppose that \(f\) is an odd function of \(x .\) Does knowing that \(\lim _{x \rightarrow 0^{+}} f(x)=3\) tell you anything about lim \(_{x \rightarrow 0} f(x) ?\) Give reasons for your answer.

Use the Intermediate Value Theorem to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations. $$ \cos x=x \quad \text {(one root). Make sure you are using radian mode.} $$

How many horizontal asymptotes can the graph of a given rational function have? Give reasons for your answer.
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