Problem 36 - Calculus for Biology and Medicine

Question

For each function, find the largest possible domain and determine the range. $$ f(x)=\frac{1}{x^{2}+1} $$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\), Range: \((0, 1]\).

Step 1: Identify the Restrictions for the Domain

For the function \( f(x) = \frac{1}{x^2 + 1} \), we need to determine the values of \( x \) for which the function is defined. The primary concern is ensuring that the denominator is not equal to zero. Since \( x^2 + 1 \geq 1 \) for all real numbers \( x \), the denominator is never zero. Therefore, there are no restrictions on \( x \).

Step 2: State the Largest Possible Domain

Given that \( x^2 + 1 \) is never zero, the function \( f(x) = \frac{1}{x^2 + 1} \) is defined for all real numbers \( x \). Thus, the largest possible domain is all real numbers, \( (-\infty, \infty) \).

Step 3: Determine the Range by Analyzing the Function

To find the range, evaluate the behavior of \( f(x) = \frac{1}{x^2 + 1} \). Note that \( x^2 \geq 0 \) implies \( x^2 + 1 \geq 1 \). This makes \( f(x) \) always less than or equal to \( 1 \). Since \( x^2 \) is non-negative and \( f(x) = \frac{1}{x^2 + 1} \) never reaches zero, it approaches \( 0 \) as \( x^2 \) becomes very large.

Step 4: Express the Range

The function values \( f(x) = \frac{1}{x^2 + 1} \) can never actually reach zero but can get arbitrarily close to it. Therefore, the range of the function is \((0, 1]\).

Key Concepts

Function Domain

The domain of a function refers to the set of all possible input values (or x-values) for which the function is defined. When determining the domain, we look for any restrictions that might prevent an x-value from being permissible. For rational functions, like the function given \[ f(x) = \frac{1}{x^2 + 1} \], we focus on ensuring the denominator is not zero because division by zero is undefined in mathematics. However, for \( x^2 + 1 \), no real number will make the denominator zero, since \( x^2 \) is always non-negative and adding 1 guarantees a positive outcome. Thus, the largest possible domain for this function includes all real numbers.

• Rational functions are expressions that are the ratio of two polynomials.
• The domain is determined by where the denominator does not equal zero.
• For \( f(x) = \frac{1}{x^2 + 1} \), the domain is all real numbers \((-\infty, \infty)\).

Function Range

The range of a function represents the set of all possible output values (or y-values). To find this, we analyze the behavior of the function across its domain. For the function \( f(x) = \frac{1}{x^2 + 1} \), the key observation is that the denominator \( x^2 + 1 \) is always greater than or equal to 1. Hence, the maximum value that \( f(x) \) can achieve is 1, which occurs when \( x = 0 \). As \( x \) moves away from zero, \( x^2 + 1 \) grows larger, making \( f(x) \) approach but never reach zero. This gives us the range as \( (0, 1] \).

• The range is the set of all possible values that the function can output.
• By analyzing the function \( f(x) = \frac{1}{x^2 + 1} \), we see that it approaches zero but reaches up to 1.
• Therefore, the range is \( (0, 1] \).

Rational Functions

Rational functions are expressions that involve ratios of polynomials, such as \( f(x) = \frac{1}{x^2 + 1} \). Understanding how they work is pivotal in calculus and algebra.Characteristics of Rational Functions:

• Rational functions are defined as the quotient of two polynomials. The numerator and denominator coefficients are both polynomials.
• The domain of a rational function is all real numbers, except where the denominator is zero.
• The range is determined by the possible outputs based on input values, considering asymptotics and intercepts.
• Graphing helps visualize asymptotes and intercepts, revealing behavior such as approach to certain lines or values.

To wrap it up, rational functions can appear complex due to their non-linear nature and undefined points, yet understanding their behavior, impacts, and constraints can help students navigate mathematical challenges effectively.