Problem 14 - Thomas Calculus

Question

At what points are the functions in Exercises 13-30 continuous? $$y=\frac{1}{(x+2)^{2}}+4$$

Step-by-Step Solution

Answer

The function is continuous for all \(x\) except \(x = -2\).

Step 1: Understanding Continuity

A function is continuous at a point if it is defined at that point, its limit exists at that point, and the limit equals the function's value at that point. For rational functions, continuity issues typically arise where the denominator is zero.

Step 2: Identifying Potential Discontinuity

Examine the denominator of the function \( y = \frac{1}{(x+2)^2} + 4 \). The potential issue for discontinuity is when the denominator \((x+2)^2\) equals zero, which occurs when \(x+2 = 0\). Solving this, we find \(x = -2\).

Step 3: Determining the Domain of Continuity

The function \( y = \frac{1}{(x+2)^2} + 4 \) is continuous for all real numbers except where the denominator is zero. Since \( (x+2)^2 eq 0 \) for all \( x eq -2 \), the function is continuous for all real numbers except \( x = -2 \).

Step 4: Stating the Result

The function \( y = \frac{1}{(x+2)^2} + 4 \) is continuous everywhere except at \( x = -2 \), where it has a discontinuity due to division by zero.

Key Concepts

Rational Functions

Rational functions are a type of function formed by the ratio of two polynomials. These functions take the general form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. They are called rational because they involve a division operation similar to a fraction.

Key characteristics of rational functions include:

• The numerator and the denominator are polynomial expressions.
• Rational functions can exhibit many familiar features like horizontal and vertical asymptotes or end behaviors.

The function \( y=\frac{1}{(x+2)^2}+4 \) is an example of a rational function, where the numerator is \(1\) and the denominator is \((x+2)^2\). This function shows additional behavior by shifting up by 4 units due to the constant at the end.

One major concern when dealing with rational functions is identifying the values that make the denominator zero, as these values usually result in discontinuities. Understanding rational functions includes determining where these breaks, or points of discontinuity, occur and how they affect the function's graph and overall continuity.

Point of Discontinuity

The point of discontinuity occurs where a function is not continuous. For rational functions like \( y=\frac{1}{(x+2)^2}+4 \), discontinuities often happen where the denominator of the function equals zero, leading to undefined values.

To find the point of discontinuity:

• Set the denominator equal to zero: \((x+2)^2 = 0\).
• Solve the equation for \(x\), which in this case gives \(x = -2\).

This means the function is not defined at \(x = -2\), causing a break in the graph. At this point, the function cannot have a real output since we cannot divide by zero.

Discontinuities significantly affect the function's behavior. They are depicted as gaps or jumps in the graph. In real-world applications, recognizing and understanding these discontinuities help predict scenarios where a function might fail or need adjustment.

Domain of Continuity

The domain of continuity refers to the set of all points where a function is continuous. For the function \( y=\frac{1}{(x+2)^2}+4 \), determining this involves identifying all the points, except where the denominator becomes zero.

For rational functions, such as \( \frac{1}{(x+2)^2} \), the domain is defined for all real numbers except:\[ \begin{align*} (x+2)^2 eq 0 & \text{ implies } x eq -2. \end{align*} \]

Thus, the domain of continuity for \( y=\frac{1}{(x+2)^2}+4 \) is all real numbers except \(x = -2\).

Understanding where a function is continuous helps in sketching accurate graphs and predicting behavior in varying conditions. It's crucial to always determine the domain first, allowing you to avoid inputs that could lead to undefined outputs or errors in calculations.