The equation y = (x-4)/(2x+1) represents a rational function. To discuss its behavior, we need to analyze the various aspects of the function, including its domain, vertical asymptotes, horizontal asymptotes, x-intercepts, y-intercepts, and any discontinuities.
1. Domain: The function is defined for all real values of x except when the denominator (2x+1) is equal to zero. Therefore, the domain of the function is all real numbers except x = -1/2.
2. Vertical Asymptotes: Vertical asymptotes occur where the denominator becomes zero. In this case, the vertical asymptote occurs at x = -1/2.
3. Horizontal Asymptotes: To determine the horizontal asymptote, we look at the degrees of the numerator and denominator. The degree of the numerator (x-4) is 1, and the degree of the denominator (2x+1) is also 1. Since the degrees are equal, the horizontal asymptote can be found by taking the ratio of the leading coefficients. In this case, the horizontal asymptote is at y = 1/2.
4. x-intercepts: To find the x-intercept, we set y = 0 and solve for x. Therefore, (x-4)/(2x+1) = 0, and the only solution to this equation is x = 4. So, the x-intercept is (4, 0).
5. y-intercept: To find the y-intercept, we set x = 0 and evaluate the function. Plugging in x = 0, we get y = -4/1 = -4. So, the y-intercept is (0, -4).
6. Discontinuities: The function has a discontinuity at x = -1/2 due to the vertical asymptote. Other than that, there are no other apparent discontinuities.
In summary, the behavior of the rational function y = (x-4)/(2x+1) includes a vertical asymptote at x = -1/2, a horizontal asymptote at y = 1/2, an x-intercept at (4, 0), and a y-intercept at (0, -4).
dicuss the behaviour of y=(x=4)/(2x+1)
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