f(x)=(1)/(x-1)

Question

f(x)=(1)/(x-1)

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Identify Domain: To understand the characteristics of the function pf(x) = (1)/(x-1), we first need to identify the domain of the function. The domain consists of all the values of x for which the function is defined. Since division by zero is undefined, we must exclude the value of x that makes the denominator zero. In this case, x - 1 = 0 when x = 1. Therefore, the domain of pf(x) is all real numbers except x = 1.Vertical Asymptote: Next, we should consider the vertical asymptote of the function. A vertical asymptote occurs at the values of x that are excluded from the domain and cause the function to approach infinity. Since x = 1 is excluded from the domain, there is a vertical asymptote at x = 1.Horizontal Asymptote: We can also look at the horizontal asymptote of the function. For rational functions, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the x-axis, or y = 0. Since the numerator is a constant (1) and the denominator is a first-degree polynomial (x-1), the degree of the numerator is less than the degree of the denominator. Therefore, there is a horizontal asymptote at y = 0.Behavior around Asymptote: Another characteristic to consider is the behavior of the function around the vertical asymptote. As x approaches 1 from the left (x < 1), the denominator (x-1) becomes negative and approaches zero, causing the function value to approach negative infinity. As x approaches 1 from the right (x > 1), the denominator (x-1) becomes positive and approaches zero, causing the function value to approach positive infinity.Continuity: Finally, we should note that the function is continuous everywhere in its domain. Since the only point of discontinuity is x = 1, which is not in the domain, the function is continuous for all other values of x.

$f(x)=\frac{1}{x-1}$

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$f(x)=\frac{1}{x-1}$