Problem 28 - Calculus: Early Transcendentals

Question

Continuity Determine the interval(s) on which the following functions are continuous. $$s(x)=\frac{x^{2}-4 x+3}{x^{2}-1}$$

Step-by-Step Solution

Answer

Answer: The function s(x) is continuous on the intervals \((-\infty, -1) \cup (-1, 1) \cup (1, \infty)\).

Step 1: Identify the denominator of the function

In this case, we are given the function: $$s(x)=\frac{x^{2}-4 x+3}{x^{2}-1}$$ The denominator of the function is: $$x^{2}-1$$

Step 2: Determine the values of x that cause the denominator to be zero

To find the values of x that cause the denominator to be zero, we solve the equation: $$x^{2}-1=0$$ This equation can be solved using factoring: $$(x-1)(x+1)=0$$ The values of x that make this equation true are x=1 and x=-1.

Step 3: Identify the intervals on which the function is continuous

We found that the function is undefined for x=1 and x=-1. Hence, we need to consider the intervals excluding these points. Therefore, we have the following intervals for the function being continuous: $$(-\infty, -1) \cup (-1, 1) \cup (1, \infty)$$ So, the function s(x) is continuous on the intervals \((-\infty, -1) \cup (-1, 1) \cup (1, \infty)\).

Key Concepts

Denominator

When dealing with rational functions, the denominator holds particular importance. It's the part of the fraction that lies below the dividing line, and essentially dictates where the function might be undefined. For the given function \(s(x) = \frac{x^2 - 4x + 3}{x^2 - 1}\), the denominator \(x^2 - 1\) plays a critical role. If this part of the function equals zero, the whole fraction is undefined because division by zero is not allowed in mathematics. This is why identifying the denominator is the first step when determining the intervals of continuity for rational functions.
Denominators are crucial because:

• They identify potential points of discontinuity.
• They determine where the function can have vertical asymptotes.
• They help in understanding the overall behavior of the function across its domain.

Hence, grasping the nature of the denominator allows us to pinpoint exactly where a function may falter in terms of defining a value.

Factoring

Factoring is a mathematical process often used to simplify expressions or solve equations. It involves breaking down a complex expression into a product of simpler factors. In our scenario with the denominator \(x^2 - 1\), factoring allows us to find the roots, which are points that can potentially disrupt the function's continuity.
In the case of \(x^2 - 1\), we can factor it as \((x-1)(x+1)\). Factoring reveals that the equation \(x^2 - 1 = 0\) has solutions at \(x = 1\) and \(x = -1\). These solutions tell us precisely where the original function is undefined.
Here are some essentials about factoring:

• It simplifies complex polynomials into more manageable expressions.
• It uncovers the roots of equations, which are critical in finding points of discontinuity.
• It is a step that makes subsequent calculations much easier by reducing polynomial complexity.

Thus, choosing to factor here is what directly lets us identify the break points in the function.

Intervals of Continuity

In calculus, determining intervals of continuity is crucial because it tells us where a function behaves "nicely" without any interruptions or breaks. For the function \(s(x) = \frac{x^2 - 4x + 3}{x^2 - 1}\), we've identified that it is undefined at \(x = 1\) and \(x = -1\). This means that these points must be excluded from intervals of continuity.
With the function being undefined or having a discontinuity at these points, the continuous intervals are identified as:

• \((-\infty, -1)\)
• \((-1, 1)\)
• \((1, \infty)\)

These intervals mean the function is continuous everywhere except at \(x = 1\) and \(x = -1\). By listing intervals separately, we're acknowledging where the function transitions smoothly across its domain without disruption. This understanding aids in various applications, such as integration and curve analysis, where knowing continuous behavior simplifies computations and explanations.