Problem 467 - Precalculus

Question

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions. $$f(x)=\frac{x}{x^{2}+5 x-36}$$

Step-by-Step Solution

Answer

Domain: all real numbers except \( x = -9 \) and \( x = 4 \); Vertical Asymptotes: \( x = -9 \), \( x = 4 \); Horizontal Asymptote: \( y = 0 \).

Step 1: Identify the Domain

The domain of a function is the set of all possible input values (x-values) that will not cause the function to be undefined. For rational functions, this generally means looking for x-values that make the denominator zero. Given the function \( f(x)= \frac{x}{x^2+5x-36} \), we need to find where \( x^2 + 5x - 36 = 0 \). We will factor the quadratic to find these values: \( (x + 9)(x - 4) = 0 \). Thus, the denominator is zero at \( x = -9 \) and \( x = 4 \). The domain is therefore all real numbers except \( x = -9 \) and \( x = 4 \).

Step 2: Find the Vertical Asymptotes

Vertical asymptotes occur where the function is undefined, due to division by zero, and the numerator is not zero. From Step 1, we identified that the function is undefined at \( x = -9 \) and \( x = 4 \). Neither of these x-values make the numerator zero. Therefore, the vertical asymptotes are \( x = -9 \) and \( x = 4 \).

Step 3: Identify the Horizontal Asymptote

The horizontal asymptote of a rational function is determined by the degrees of the polynomials in the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the x-axis (\( y = 0 \)) is the horizontal asymptote. Here, the numerator \( x \) is degree 1, and the denominator \( x^2 + 5x - 36 \) is degree 2. Since the numerator's degree is less than the denominator's degree, the horizontal asymptote is \( y = 0 \).

Key Concepts

Domain of a Function

The domain of a function tells you which x-values are allowed. For rational functions, the big concern is division by zero. If the denominator becomes zero for any x-value, that x-value is not part of the domain. To figure out where this happens, you set the denominator equal to zero and solve for x. Let's look at the given function: \[ f(x) = \frac{x}{x^2 + 5x - 36} \]Here, you solve the equation: \[ x^2 + 5x - 36 = 0 \]By factoring, you find: \[ (x + 9)(x - 4) = 0 \]This shows up when:

• \( x = -9 \)
• \( x = 4 \)

These x-values make the function undefined, as they lead to division by zero. Thus, the domain includes all real numbers except \( x = -9 \) and \( x = 4 \). This means to avoid those numbers when you pick x-values for the function.

Vertical Asymptotes

Vertical asymptotes are strange parts of rational functions where the graph shoots up to infinity or down to negative infinity. These occur at x-values which make the function undefined, much like when finding the domain. So, vertical asymptotes happen when the denominator is zero and the numerator isn’t zero at the same place. In our function:\[ f(x) = \frac{x}{x^2 + 5x - 36} \]We already know from the domain discussion that:

• \( x = -9 \)
• \( x = 4 \)

Both make \( x^2 + 5x - 36 \) zero. Importantly, these do not make the numerator zero.Thus, the vertical asymptotes are at \( x = -9 \) and \( x = 4 \). When you draw the graph, it will get very steep around these points but never actually touch them. This behavior is what defines vertical asymptotes.

Horizontal Asymptotes

Horizontal asymptotes reflect the behavior of a graph as x moves towards infinity or negative infinity. These are common in rational functions and can be calculated by comparing the degrees (highest powers) of the numerator and the denominator.Consider our function:\[ f(x) = \frac{x}{x^2 + 5x - 36} \]This has:

• Numerator of degree 1 (since the highest power of x is \( x^1 \))
• Denominator of degree 2 (since the highest power of x is \( x^2 \))

When the denominator’s degree is higher, like here, the horizontal asymptote is at \( y = 0 \). This tells you that as x goes far to the right or left, the function approaches zero. It doesn't mean the graph will touch this line, but it will get closer and closer indefinitely. Always remember, horizontal asymptotes deal with where the graph heads as x becomes very large or very small, unlike vertical asymptotes which address certain x-values.