Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = 2x / (x² − 1)

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To determine the domain of the function \( y = \frac{2x}{x^2 - 1} \), identify the values of \( x \) that make the denominator zero, as these are the points where the function is undefined. Set \( x^2 - 1 = 0 \) and solve for \( x \).

Solve the equation \( x^2 - 1 = 0 \) by factoring it as \( (x - 1)(x + 1) = 0 \). This gives the solutions \( x = 1 \) and \( x = -1 \). Therefore, the domain of the function is all real numbers except \( x = 1 \) and \( x = -1 \).

Next, find the vertical asymptotes by examining the behavior of the function as \( x \) approaches the values that make the denominator zero. Calculate the limits \( \lim_{x \to 1^+} \frac{2x}{x^2 - 1} \), \( \lim_{x \to 1^-} \frac{2x}{x^2 - 1} \), \( \lim_{x \to -1^+} \frac{2x}{x^2 - 1} \), and \( \lim_{x \to -1^-} \frac{2x}{x^2 - 1} \).

To find horizontal asymptotes, consider the limit of the function as \( x \) approaches infinity. Calculate \( \lim_{x \to \infty} \frac{2x}{x^2 - 1} \) and \( \lim_{x \to -\infty} \frac{2x}{x^2 - 1} \).

Analyze the results of these limits: if the limit approaches a finite number as \( x \to \infty \) or \( x \to -\infty \), that number is the horizontal asymptote. If the limits as \( x \to 1 \) or \( x \to -1 \) approach infinity, these are vertical asymptotes.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions like y = 2x / (x² − 1), the domain excludes values that make the denominator zero, as division by zero is undefined. In this case, x² − 1 = 0 when x = ±1, so the domain is all real numbers except x = ±1.

Limits and Asymptotes

Limits help determine the behavior of a function as it approaches a particular point or infinity. Vertical asymptotes occur where the function approaches infinity, typically where the denominator is zero. Horizontal asymptotes describe the function's behavior as x approaches infinity. For y = 2x / (x² − 1), vertical asymptotes are at x = ±1, and the horizontal asymptote is y = 0, as the degree of the denominator is greater than the numerator.

Rational Functions

A rational function is a ratio of two polynomials. The behavior of rational functions is often analyzed by examining their domains, asymptotes, and intercepts. Understanding the structure of the numerator and denominator helps predict the function's graph. For y = 2x / (x² − 1), the numerator is linear, and the denominator is quadratic, influencing the function's asymptotic behavior and domain restrictions.

Domains and AsymptotesDetermine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.y = 2x / (x² − 1)