Problem 17 - College Algebra with Corequisite Support

Question

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions. $$ f(x)=\frac{x+5}{x^{2}-25} $$

Step-by-Step Solution

Answer

Domain: \( x \neq \pm 5 \); Vertical asymptotes: \( x = \pm 5 \); Horizontal asymptote: \( y = 0 \).

Step 1: Identify Domain

To find the domain of the function \( f(x) = \frac{x+5}{x^2-25} \), determine for which values of \( x \) the denominator is not zero. Set the denominator equal to zero: \( x^2 - 25 = 0 \). This factors as \( (x-5)(x+5) = 0 \) giving solutions \( x = 5 \) and \( x = -5 \). The domain is all real numbers except \( x = 5 \) and \( x = -5 \).

Step 2: Find Vertical Asymptotes

Vertical asymptotes occur where the function is undefined and there is no cancellation between the numerator and the denominator. Here, the points \( x = 5 \) and \( x = -5 \) both make the denominator zero but not the numerator, hence both are vertical asymptotes.

Step 3: Determine Horizontal Asymptotes

To find horizontal asymptotes, consider the behavior as \( x \to \pm \infty \). The degree of the numerator \( (x+5) \) is 1 and the degree of the denominator \( (x^2-25) \) is 2. Since the degree of the denominator is greater than the numerator, the horizontal asymptote is \( y = 0 \).

Key Concepts

Domain of a function

When talking about the domain of a function, it refers to all the possible input values (usually denoted as \( x \)) that the function can accept without resulting in any "mathematical mishaps" like division by zero or taking the square root of a negative number. For rational functions, such as \( f(x) = \frac{x+5}{x^2-25} \), these mishaps can occur when the denominator equals zero.
\( x^2 - 25 = 0 \) simplifies to \( (x-5)(x+5) = 0 \), giving us \( x=5 \) and \( x=-5 \) as problematic values. Therefore, the domain of \( f(x) \) includes all real numbers except \( x = 5 \) and \( x = -5 \). This ensures the function outputs are meaningful and avoid undefined results.

Vertical asymptotes

Vertical asymptotes are lines that a graph approaches but never actually touches or crosses. They occur in rational functions where the function is undefined due to division by zero. In our example function \( f(x)=\frac{x+5}{x^2-25} \), we've already determined that the function is undefined at \( x = 5 \) and \( x = -5 \). Since there is no cancellation between the numerator \( x+5 \) and the denominator \( (x-5)(x+5) \), both \( x = 5 \) and \( x = -5 \) are vertical asymptotes. The graph will rise or fall steeply as it approaches these lines, creating a gap in the graph at these values.

Horizontal asymptotes

Horizontal asymptotes tell us about the behavior of a function as \( x \) approaches positive or negative infinity. They represent the value that the function outputs get closer to but do not necessarily reach. For the function \( f(x)=\frac{x+5}{x^2-25} \), consider the degrees of the polynomial in the numerator and the denominator.
The degree of the numerator is 1 (from \( x+5 \)) and the degree of the denominator is 2 (from \( x^2-25 \)). When the denominator's degree is greater than the numerator's, it indicates that the outputs of the function approach 0 as \( x \) gets very large positively or negatively. Hence, the horizontal asymptote for \( f(x) \) is \( y = 0 \). This means that for very large values or very negative values of \( x \), the value of \( f(x) \) will get arbitrarily close to 0.