Sketch the graphs of the rational functions in Exercises 53–60.
y = (x² + 1) / x

검증된 단계별 안내

Identify the rational function: \( y = \frac{x^2 + 1}{x} \). This function is a rational function because it is the ratio of two polynomials.

Determine the domain of the function. The denominator \( x \) cannot be zero, so the domain is all real numbers except \( x = 0 \).

Find the vertical asymptote by setting the denominator equal to zero. Since \( x = 0 \) makes the denominator zero, there is a vertical asymptote at \( x = 0 \).

Simplify the function if possible. Here, \( y = \frac{x^2 + 1}{x} = x + \frac{1}{x} \). This helps in understanding the behavior of the function as \( x \to \infty \) and \( x \to -\infty \).

Analyze the behavior of the function as \( x \to \infty \) and \( x \to -\infty \). As \( x \to \infty \), \( y \approx x \), and as \( x \to -\infty \), \( y \approx x \). This suggests that the graph will resemble the line \( y = x \) for large values of \( |x| \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

주요 개념

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

Rational Functions

A rational function is a ratio of two polynomials, where the numerator and denominator are polynomial expressions. Understanding the behavior of rational functions involves analyzing their asymptotes, intercepts, and overall shape. In this case, the function y = (x^2 + 1) / x is a rational function with a polynomial numerator and a linear denominator.

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator equals zero, causing the function to approach infinity. For y = (x^2 + 1) / x, the vertical asymptote is at x = 0, since the denominator becomes zero at this point. This asymptote divides the graph into separate regions, influencing the function's behavior near x = 0.

End Behavior

End behavior describes how a function behaves as x approaches positive or negative infinity. For rational functions, this is often determined by the degrees of the numerator and denominator. In y = (x^2 + 1) / x, as x approaches infinity, the function behaves like y = x, since the highest degree term in the numerator dominates, indicating a linear growth.

Sketch the graphs of the rational functions in Exercises 53–60.y = (x2 + 1) / x