Problem 464 - Precalculus

Question

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions. $$f(x)=\frac{4}{x-1}$$

Step-by-Step Solution

Answer

Domain: \( x \neq 1 \); Vertical asymptote: \( x=1 \); Horizontal asymptote: \( y=0 \).

Step 1: Determine the Domain

The domain of a function is the set of all possible input values (x-values) that will not lead to undefined expressions in the function. Since the denominator in the function \( f(x)=\frac{4}{x-1} \) cannot be zero (as division by zero is undefined), we solve \( x-1=0 \) to find the value that \(x\) must not equal. Solving this gives \( x=1 \). Therefore, the domain is all real numbers except \( x=1 \). Mathematically, the domain is \( x \in \mathbb{R}, x eq 1 \).

Step 2: Find the Vertical Asymptotes

Vertical asymptotes occur where a function grows indefinitely, typically where the denominator is zero. To find vertical asymptotes for \( f(x) = \frac{4}{x-1} \), we look for values that make the denominator zero as we found in Step 1. Setting \( x-1=0 \) gives \( x=1 \). Therefore, there is a vertical asymptote at \( x=1 \).

Step 3: Find the Horizontal Asymptotes

Horizontal asymptotes are found by considering behavior as \( x \) approaches infinity or negative infinity. For \( f(x)=\frac{4}{x-1} \), as \( x \to \infty \), \( f(x)\) approaches \( 0 \). Similarly, as \( x \to -\infty \), \( f(x)\) also approaches \( 0 \). Thus, there is a horizontal asymptote at \( y=0 \).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial, as it tells us the complete set of possible input values (\( x \)-values) for which the function is defined and yields real numbers. For the function \( f(x) = \frac{4}{x-1} \), we need to ensure that the denominator is never zero, since division by zero is undefined and leads to mathematical inconsistencies. This means we need to solve the equation \( x - 1 = 0 \) to identify what to exclude from the domain.

By solving, we find that \( x eq 1 \). Thus, the function is defined for all real \( x \) except \( x = 1 \).
This can be written in interval notation as:

• \( (-\infty, 1) \cup (1, \infty) \)

In simple terms, the domain is all real numbers, except when \( x \) is 1. Grasping this concept helps avoid undefined expressions and ensures smooth evaluations of the function.

Vertical Asymptotes

Vertical asymptotes are fascinating aspects of rational functions. These asymptotes represent values of \( x \) where the function experiences infinite growth or decay. Simply put, vertical asymptotes occur at points where a factor in the denominator becomes zero, and as a result, the function value grows towards infinity or negative infinity.

For the function \( f(x) = \frac{4}{x-1} \), we previously determined the troublesome value in the denominator by setting \( x - 1 = 0 \). This occurs precisely when \( x = 1 \). As \( x \) approaches 1 from either side, \( f(x) \) tends to infinity, creating a vertical asymptote.
Therefore, the vertical asymptote for this function is located at:

• \( x = 1 \)

Vertical asymptotes graphically appear as vertical lines on a graph where the curve of the function approaches but never touches the line. Recognizing these points provides insights into the behavior and the nature of rational functions.

Horizontal Asymptotes

Horizontal asymptotes indicate the value that a function approaches as \( x \) becomes very large, either positively or negatively. Essentially, they offer a long-term prediction of the function's value as it stretches out towards infinity or negative infinity.

For the rational function \( f(x) = \frac{4}{x-1} \), as \( x \) increases towards infinity, the fraction aspect of the function's expression diminishes in magnitude because the ratio of a constant (4) over an exceedingly large number (\( x-1 \)) produces a very small number. Therefore, \( f(x) \) lands close to zero.
Here's what we observe:

• As \( x \to \infty \), \( f(x) \to 0 \)
• As \( x \to -\infty \), \( f(x) \to 0 \)

Thus, the horizontal asymptote is given by:

• \( y = 0 \)

Horizontal asymptotes show up on graphs as horizontal lines. Unlike vertical asymptotes, functions might cross these lines at certain points. However, they provide a general boundary as \( x \) continues indefinitely in either direction. Identifying horizontal asymptotes enhances comprehension of the overall behavior of the function over extended ranges.