Problem 68 - Brief Applied Calculus

Question

Determine whether each function is continuous or discontinuous. If discontinuous, state where it is discontinuous. $$ f(x)=\frac{x}{(x+7)(x-2)} $$

Step-by-Step Solution

Answer

The function is discontinuous at \( x = -7 \) and \( x = 2 \).

Step 1: Identify Potential Discontinuities

A function is discontinuous where its denominator is equal to zero, as these are the points where the function is undefined. For the function \( f(x) = \frac{x}{(x+7)(x-2)} \), set the denominator \((x+7)(x-2)\) equal to zero to find these points. Solve the equations:1. \(x+7 = 0 \) gives \( x = -7 \).2. \(x-2 = 0 \) gives \( x = 2 \).Thus, potential points of discontinuity are at \(x = -7\) and \(x = 2\).

Step 2: Evaluate Continuity Separately

For each potential discontinuity, evaluate the overall function without delving into unnecessary details:- At \( x = -7 \), the function \( f(x) = \frac{x}{(x+7)(x-2)} \) is undefined, indicating a discontinuity.- At \( x = 2 \), the function \( f(x) = \frac{x}{(x+7)(x-2)} \) is undefined, indicating another discontinuity. Thus, the function is discontinuous at both \( x = -7 \) and \( x = 2 \).

Key Concepts

Discontinuous Points

When dealing with functions, discontinuous points are where a function does not behave in a smooth and continuous manner. These points occur where the denominator of a function becomes zero, making the function undefined. In the example of the function \( f(x) = \frac{x}{(x+7)(x-2)} \), the discontinuous points occur at values of \( x \) that make the denominator zero.To find these points, set \((x+7)(x-2) = 0\). Solve for \( x \):

• \( x+7 = 0 \) gives \( x = -7 \)
• \( x-2 = 0 \) gives \( x = 2 \)

These solutions, \( x = -7 \) and \( x = 2 \), are where the function is discontinuous. This indicates that there is a break or hole in the graph of the function at these points. Always pinpoint these spots as they help us understand where the function is not defined.

Evaluating Functions

Evaluating functions is a process where we substitute different values of \( x \) into the function to determine the corresponding output value. It helps us to understand the behavior of the function across its domain. When examining functions for continuity, it's important to test the boundaries at potential discontinuous points to verify the smoothness of the function.For the function \( f(x) = \frac{x}{(x+7)(x-2)} \), we specifically look at the values around the discovered discontinuous points, \( x = -7 \) and \( x = 2 \). However, it's essential to note that at these exact values the function cannot be evaluated due to the undefined nature. But just before and after these points, the function can be checked for values to see if there's any continuity leading up to, or away from, these bounds.Evaluating functions in this manner shows how a function behaves in an overall sense, giving us insight into its continuous or discontinuous nature.

Undefined Expressions

Undefined expressions occur in functions when operations like division include a zero in the denominator, making the entire expression indeterminate. These are critical points at which a function cannot be evaluated and are often the root of discontinuous points.In the function \( f(x) = \frac{x}{(x+7)(x-2)} \), we find undefined expressions by factoring the denominator and setting it to zero. The expressions \((x+7)(x-2)\) guide us to find the values of \( x = -7 \) and \( x = 2 \), where the expressions become zero.Recognizing and acknowledging undefined expressions helps us grasp the structure of the function's domain and limitations, addressing specifically where and why the function cannot be evaluated.