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Chapter 3: Problem 465

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

Short Answer

Expert verified
Domain: \((-\infty, -\frac{2}{5}) \cup (-\frac{2}{5}, \infty)\); Vertical asymptote: \(x = -\frac{2}{5}\); Horizontal asymptote: \(y = 0\).

Step by step solution

01

Find the Domain

The domain of a function is the set of all possible input values (x-values) that will not produce division by zero or any undefined expressions. For the function \( f(x) = \frac{2}{5x + 2} \), the denominator is \( 5x + 2 \). Set this equal to zero to find the restricted values: \[ 5x + 2 = 0 \]Solve for \( x \):\[ 5x = -2 \]\[ x = -\frac{2}{5} \]Thus, the domain is all real numbers except \( x = -\frac{2}{5} \) or in interval notation, \( (-\infty, -\frac{2}{5}) \cup (-\frac{2}{5}, \infty) \).
02

Find the Vertical Asymptotes

Vertical asymptotes occur at values of \( x \) where the function tends toward infinity, often when the denominator is zero. From Step 1, we determined that the denominator \( 5x + 2 = 0 \) at \( x = -\frac{2}{5} \). Therefore, there is a vertical asymptote at \( x = -\frac{2}{5} \).
03

Find the Horizontal Asymptotes

Horizontal asymptotes are found by analyzing the behavior of the function as \( x \) approaches infinity. For a rational function \( \frac{a}{bx+c} \), the horizontal asymptote is at \( y = 0 \) if the degree of the polynomial in the denominator is greater than the degree of the numerator:- The degree of the numerator \( 2 \) is 0 since it's a constant function.- The degree of the denominator \( 5x + 2 \) is 1 since it's a linear expression.Since the degree of the denominator is greater, \( f(x) \) approaches 0 as \( x \to \pm \infty \). Thus, the horizontal asymptote is \( y = 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Functions
A rational function is a ratio of two polynomials, where the numerator and the denominator are polynomial expressions. In our exercise, the function given is \[ f(x) = \frac{2}{5x + 2} \]Here:
  • The numerator is "2", a constant term, which is a polynomial of degree 0.
  • The denominator is "5x + 2", a linear polynomial of degree 1.
Rational functions can create various behaviors or shapes on a graph. These behaviors include the influence of the polynomials' degrees on the existence and position of asymptotes. Since the variable "x" is only present in the denominator, it indicates the potential for division by zero, an important characteristic in analyzing the function.
Vertical Asymptotes
Vertical asymptotes are lines that the graph of the function approaches but never touches. They occur at the x-values where the denominator equals zero, as the function tends toward infinity or negative infinity at these points. Let鈥檚 recall our equation: \[ f(x)=\frac{2}{5x+2} \]
  • To find where the vertical asymptotes are, we set the denominator equal to zero because division by zero is undefined.
  • For \( 5x + 2 = 0 \), solve for \( x \): \[ 5x = -2 \] \[ x = -\frac{2}{5} \]
Thus, \( x = -\frac{2}{5} \) is the location of the vertical asymptote. This is a crucial point as the function behaves erratically at this value.
Horizontal Asymptotes
Horizontal asymptotes describe the behavior of a function as the input \( x \) becomes very large or very small, approaching infinity. They indicate the line which the function values approach but may never reach.To analyze these for our rational function \( f(x) = \frac{2}{5x + 2} \), we compare the degrees of the numerator and denominator:
  • The degree of the numerator (2) is 0, because it's a constant.
  • The degree of the denominator (5x + 2) is 1, making it a linear expression.
When the degree of the denominator is greater than that of the numerator, as is the case here, the horizontal asymptote is at \( y = 0 \). This means as \( x \rightarrow \pm \infty \), the function value gets closer and closer to zero, highlighting the asymptotic behavior.
Function Analysis
Analyzing a function like \( f(x)=\frac{2}{5x+2} \) involves understanding its behavior across all x-values. This includes determining where the function is defined, the intervals where it is increasing or decreasing, and how it behaves near its asymptotes:
  • The domain is all real numbers except \( x = -\frac{2}{5} \), due to the zero in the denominator.
  • The vertical asymptote at \( x = -\frac{2}{5} \) suggests that the function will approach positive or negative infinity there, without ever actually taking a value.
  • Because there is a horizontal asymptote at \( y = 0 \), as \( x \) grows very large or very small, the output values will inch towards zero.
Overall, function analysis enables us to visualize how these mathematical expressions translate into graph forms, predicting the shape of the curve and behavior at critical points.

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