Problem 35 - Calculus for Biology and Medicine

Question

For each function, find the largest possible domain and determine the range. $$ \text { 35. } f(x)=\frac{x-2}{x^{2}-9} $$

Step-by-Step Solution

Answer

Domain: \((-\infty, -3) \cup (-3, 3) \cup (3, \infty)\), Range: \((-\infty, 0) \cup (0, \infty)\).

Step 1: Identify the Function Restrictions

The function given is \( f(x) = \frac{x-2}{x^2-9} \). To find the largest possible domain, we need to determine where the function is undefined. A fraction is undefined when its denominator is zero. So, we set the denominator equal to zero and solve for \( x \): \[ x^2 - 9 = 0 \] This can be factored: \[ (x - 3)(x + 3) = 0 \] Solving these gives \( x = 3 \) and \( x = -3 \). Hence, the function is undefined at \( x = 3 \) and \( x = -3 \).

Step 2: Determine the Largest Domain

Since the function is only undefined at \( x = 3 \) and \( x = -3 \), these values must be excluded from the domain. Therefore, the largest possible domain is all real numbers except \( 3 \) and \( -3 \): \[ \text{Domain: } (-\infty, -3) \cup (-3, 3) \cup (3, \infty) \]

Step 3: Calculate Limits at Asymptotes

To find the range, we must consider the behavior of the function as it approaches the points where it is undefined. As \( x \to 3 \) or \( x \to -3 \), the values of \( f(x) \) approach \( \pm \infty \) due to the vertical asymptotes at these points.

Step 4: Analyze Function Behavior

The function behaves normally outside of its asymptotes. To understand the full range, analyze the behavior as \( x \to \infty \) and \( x \to -\infty \): - For \( x \to \infty \) or \( x \to -\infty \), \( f(x) \approx \frac{x}{x^2} = \frac{1}{x} \to 0 \).The function doesn't actually reach zero, but it gets arbitrarily close, both from above and below zero.

Step 5: Deduce the Range

Considering the asymptotic behavior and continuity over intervals excluding \(-3\) and \(3\), the range spans all real numbers except zero, since the function asymptotically approaches zero but never crosses it. Therefore, the range is: \[ \text{Range: } (-\infty, 0) \cup (0, \infty) \]

Key Concepts

Domain and Range

In calculus, understanding the domain and range of a function is essential for grasping its behavior. The **domain** of a function refers to the set of all possible input values (or 'x' values) for which the function is defined. For the given function \[ f(x) = \frac{x-2}{x^2-9} \],we need to check where the function is undefined to find the largest possible domain.
The fraction becomes undefined if the denominator equals zero. To solve for these, set \[ x^2 - 9 = 0 \]. Factoring gives \[ (x - 3)(x + 3) = 0 \],leading to solutions at \( x = 3 \) and \( x = -3 \). Thus, the domain includes all real numbers except these points:

• Domain: \((-\infty, -3) \cup (-3, 3) \cup (3, \infty)\)

Next, the **range** of the function signifies all possible output values. As \( x \to 3 \) or \( x \to -3 \),the function diverges to \( \pm \infty \) due to vertical asymptotes. Analyzing behavior as \( x \to \pm \infty \), we find \( f(x) \approx \frac{1}{x} \), which approaches zero but never attains it.
The range includes all real values except zero:

• Range: \((-\infty, 0) \cup (0, \infty)\)

Asymptotes

Asymptotes are lines that a graph approaches but never touches or crosses. In the function \( f(x) = \frac{x-2}{x^2-9} \), consider vertical asymptotes, which occur where the denominator equals zero. Setting \( x^2 - 9 = 0 \), we find solutions at \( x = 3 \) and \( x = -3 \). These represent vertical asymptotes since the function tends toward infinity.

• Vertical Asymptotes: \( x = 3 \) and \( x = -3 \)

Horizontal asymptotes are determined by evaluating the limit as \( x \to \infty \) or \( x \to -\infty \). In this case, the function behaves as \( f(x) \approx \frac{1}{x} \), indicating an asymptote at the x-axis, or \( y = 0 \).
Remember:

• Horizontal Asymptote: \( y = 0 \)

This diagnosis helps in predicting the behavior of rational functions.

Function Behavior

The behavior of a function involves how it behaves around asymptotes and extreme values. For a rational function like \( f(x) = \frac{x-2}{x^2-9} \), we focus on its approach toward infinity. As \( x \to 3 \) or \( x \to -3 \), the values of the function \( f(x) \) increase or decrease without bound, yielding \( \pm \infty \).
These are the vertical asymptotes, indicating a division in the graph where the function is undefined.When considering values far from these asymptotes (as \( x \to \pm \infty \)), the function transforms into \( f(x) \approx \frac{1}{x} \). This means it gradually approaches zero, hinting at a horizontal asymptote.
Analyzing this behavior gives us a better view of how the function graph stretches infinitely far from the origin without ever hitting a fixed ceiling or floor.

• For \( x \to 3 \) or \( x \to -3 \): Values \( \to \pm \infty \)
• For \( x \to \pm \infty \): Values approach zero (horizontal asymptote)

Rational Functions

Rational functions are essential in calculus as they provide a deeper understanding of polynomial divisions. These functions, represented by the ratio of two polynomials, have intriguing properties, such as asymptotes and distinct behaviors.
The function \( f(x) = \frac{x-2}{x^2-9} \), is a quintessential rational function, derived by dividing the polynomial \( x-2 \) by \( x^2-9 \). Unlike polynomials, these can become undefined at certain points where the denominator equals zero (as you've learned, \( x = 3 \) and \( x = -3 \) here).
Aside from vertical and horizontal asymptotes, rational functions can exhibit oblique asymptotes, although not present in our current analysis.
Rational functions offer unique insights:

• They model behaviors not found in simple polynomials
• They help study limits, continuity, and discontinuity effectively
• They are used across diverse fields, from engineering to economics

Mastering rational functions broadens problem-solving abilities and comprehension of complex mathematical concepts.