Problem 35 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Domain Restrictions

The function is a rational function, meaning that it is in the form of a fraction. The denominator cannot be zero as this would make the function undefined. The denominator is \(x^2 - 9\), so we set it not equal to zero: \(x^2 - 9 eq 0\).

Step 2: Solve for Domain

To find when \(x^2 - 9 = 0\), we solve the equation: \(x^2 = 9\). Taking the square root of both sides gives \(x = 3\) and \(x = -3\). Therefore, the domain excludes these values: \(x eq 3\) and \(x eq -3\). The largest possible domain is all real numbers except 3 and -3.

Step 3: Definition of Domain in Interval Notation

We represent the domain in interval notation as \((-\infty, -3) \cup (-3, 3) \cup (3, \infty)\).

Step 4: Analyze the Range of the Function

To find the range, we analyze the behavior of \(f(x)\) as \(x\) approaches the values that cause the denominator to zero (near \(x = 3\) and \(x = -3\)) and as \(x\) approaches positive and negative infinity. The function can take any real number except for the values added as horizontal asymptotes.

Step 5: Find Limit Behavior

As \(x\) approaches \(3\) or \(-3\), the values of \(f(x)\) approach infinity. Hence, these points guide the vertical asymptotes, while the overall limits as \(x\) approach positive and negative infinity should guide the evaluation of a horizontal asymptote. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).

Step 6: Define the Range in Interval Notation

Since \(f(x)\) can approach any value except the horizontal asymptote \(y = 0\), the range of the function in interval notation is \((-infty, 0) \cup (0, infty)\).

Key Concepts

Exploring Rational Functions

A rational function is a type of function that is expressed as a ratio of two polynomials. This means it will generally take the form \( \frac{p(x)}{q(x)} \), where both \(p(x)\) and \(q(x)\) are polynomials. When you see a rational function, consider it as a fraction where some values of \(x\) might make the denominator zero. These values make the function undefined because dividing by zero is not possible in mathematics. This is a crucial point to watch out for when dealing with rational functions. Rational functions are fascinating since they often come with distinct features, such as asymptotes, which are lines that the function approaches but never actually touches. These functions often have more complex behaviors than polynomials or linear functions, especially around their asymptotes.

Domain and Range

The domain of a function refers to all possible input values \(x\) for which the function is defined. In simpler terms, it's the complete list of all values that \(x\) can be that will enable \(f(x)\) to return a valid output. For rational functions, the main concern is ensuring that the denominator never equals zero, as this would lead to undefined values. Thus, identifying the domain usually involves solving an equation where the denominator is set not equal to zero.The range, on the other hand, describes the possible output values a function can produce. When considering the range of rational functions, it's essential to evaluate the behavior near vertical asymptotes and as \(x\) approaches infinity. Often, the range may exclude certain values, particularly if a horizontal asymptote exists, preventing the function from achieving those outputs.

Understanding Interval Notation

Interval notation is a straightforward and efficient way to express the domain and range of functions. Instead of listing out possible values, intervals succinctly describe a range of numbers. For instance, the interval \((-\infty, -3)\) signifies all values less than \(-3\), while \((-3, 3)\) represents values between \(-3\) and \(3\). Brackets and parentheses in interval notation provide critical information:

• Parentheses \(( )\) denote that an endpoint is not included in the interval.
• Brackets \([ ]\) show that an endpoint is included.
• The union symbol \(\cup\) connects multiple intervals, indicating a combination of distinct segments.

Understanding these symbols is important for correctly interpreting interval notation in the context of a function's domain or range. This simple notation helps in clearly stating what values \(x\) can take without ambiguity, making it essential in calculus and function analysis.