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(x^2+1)/(x+1)

4 min read Jun 17, 2024
(x^2+1)/(x+1)

Exploring the Expression (x^2 + 1)/(x + 1)

The expression (x^2 + 1)/(x + 1) represents a rational function, a function defined as the ratio of two polynomials. Let's delve into its properties and characteristics:

Domain and Asymptotes

  • Domain: The domain of this function includes all real numbers except for x = -1. This is because the denominator becomes zero at x = -1, leading to an undefined result.

  • Vertical Asymptote: Due to the restriction in the domain, the expression has a vertical asymptote at x = -1. This means the graph of the function approaches infinity as x gets closer to -1 from either side.

  • Horizontal Asymptote: We can determine the horizontal asymptote by analyzing the degrees of the numerator and denominator polynomials:

    • The numerator has a degree of 2 (x^2).
    • The denominator has a degree of 1 (x).
    • Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, the function exhibits slant asymptote behavior.

Slant Asymptote

To find the slant asymptote, we perform long division:

        x - 1 
    x + 1 | x^2 + 0x + 1
          -(x^2 + x)
          -------
               -x + 1
               -(-x - 1)
               -------
                     2

The quotient, x - 1, represents the equation of the slant asymptote. This means the graph of the function approaches the line y = x - 1 as x approaches positive or negative infinity.

Other Properties

  • Intercepts:

    • x-intercept: The function has no x-intercepts. This is because the numerator (x^2 + 1) is always positive, and the function can never equal zero.
    • y-intercept: Setting x = 0, we get (0^2 + 1)/(0 + 1) = 1. So, the y-intercept is at (0, 1).

  • Symmetry: The function is not symmetric about the y-axis or the origin.

Graphing the Function

Combining the information about domain, asymptotes, intercepts, and symmetry, we can sketch a rough graph of the function (x^2 + 1)/(x + 1):

  • The graph will have a vertical asymptote at x = -1.
  • It will have a slant asymptote at y = x - 1.
  • It will pass through the point (0, 1).
  • The graph will not intersect the x-axis.

The graph will show a curve that approaches the vertical and slant asymptotes as x approaches their respective values.

Conclusion

The expression (x^2 + 1)/(x + 1) represents a rational function with a vertical asymptote, a slant asymptote, and a y-intercept. Its domain excludes x = -1, and it has no x-intercepts. Understanding these properties allows us to visualize and analyze the behavior of the function.

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