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

Problem

ƒ(x)=x/(x−1)

Solution

1. Identify the domain by finding where the denominator is zero. Since x−1=0 when x=1 the domain is all real numbers except x=1

2. Find the vertical asymptote by setting the denominator to zero. There is a vertical asymptote at x=1

3. Find the horizontal asymptote by comparing the degrees of the numerator and denominator. Since both are degree 1, the asymptote is the ratio of the leading coefficients: y=1/1 which simplifies to y=1

4. Determine the intercepts by evaluating the function at zero and setting the function to zero. The y-intercept is ƒ(0)=0/(0−1)=0 The x-intercept is found by setting the numerator x=0 Thus, the graph passes through the origin (0,0)

5. Analyze the behavior near the vertical asymptote. As x→1 ƒ(x)→∞ As x→1 ƒ(x)→−∞

6. Sketch the graph using the asymptotes x=1 and y=1 the intercept (0,0) and additional points such as (2,2)

Final Answer

ƒ(x)=x/(x−1)

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