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Chapter 2: Problem 4

Determine the vertical asymptote(s) of each function. If none exists, state that fact. $$ f(x)=\frac{3 x}{x^{2}-9} $$

Short Answer

Expert verified
The function has vertical asymptotes at \( x = 3 \) and \( x = -3 \).

Step by step solution

01

Identify Denominator

To find vertical asymptotes, we first need to find where the denominator of the function is equal to zero. For the function \( f(x) = \frac{3x}{x^2 - 9} \), the denominator is \( x^2 - 9 \).
02

Solve for Zeroes

Set the denominator equal to zero and solve for \( x \). This gives \( x^2 - 9 = 0 \). Factor this expression to get \( (x - 3)(x + 3) = 0 \). Solving for \( x \), we get \( x = 3 \) and \( x = -3 \).
03

Confirm Vertical Asymptotes

Since the function has no common factors in the numerator and denominator, the values \( x = 3 \) and \( x = -3 \) correspond to points where the function is not defined, thus confirming the vertical asymptotes at these values.

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Key Concepts

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

Understanding Rational Functions
Rational functions are a type of function represented by the ratio of two polynomials. This means a rational function has a form similar to \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are both polynomials. It's important to note that the denominator \( Q(x) \) should not be equal to zero, since division by zero is undefined in mathematics.

When analyzing rational functions, remember that:
  • The numerator, \( P(x) \), affects the function's behavior, such as intercepts and overall shape.
  • The denominator, \( Q(x) \), determines where the function is undefined, and is crucial in finding vertical asymptotes.
Mastering rational functions will help you understand how many types of functions work and is a key foundation for calculus.
Breaking Down Function Analysis
Function analysis involves examining different aspects of a function to understand its behavior. This includes identifying intercepts, asymptotes, and intervals of increase or decrease. One fundamental process in analyzing rational functions is to check where the function breaks or changes.
  • Zeros of the Function: These are points where the numerator equals zero, giving us the x-intercepts of the graph.
  • Undefined Points: Determine where the denominator \( Q(x) \) is zero, since the function does not exist at these points.
  • Simplifying the Function: Always simplify the function by canceling out common factors from the numerator and the denominator, if possible.
The aim is to gather as much information as possible about the function's graph to predict its shape and behavior accurately.
Determining Asymptotes
Asymptotes are critical lines that a graph approaches, but never touches. These include vertical, horizontal, and oblique asymptotes. Here, we'll focus on vertical asymptotes, which occur when the denominator equals zero.
  • Setting the Denominator to Zero: To find vertical asymptotes, set \( Q(x) = 0 \) and solve for \( x \). In our example, \( x^2 - 9 = 0 \), factors to \( (x - 3)(x + 3) = 0 \), resulting in \( x = 3 \) and \( x = -3 \).
  • Checking Function Simplification: Be sure that these solutions do not cancel out with terms in the numerator; if they do, they may represent a hole instead, not an asymptote.
  • Graph Behavior Near Asymptotes: As \( x \) approaches these values, the function will increase or decrease without bound, appearing as steep vertical lines on the graph.
Remember, recognizing vertical asymptotes in rational functions can greatly assist in graphing and predicting their behavior effectively.

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