Graph x/(x+1)

Problem

ƒ(x)=x/(x+1)

Solution

1. Identify the domain by finding where the denominator is zero.

x+1=0

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 evaluating the limit as x approaches infinity.

(lim_x→∞)(x/(x+1))=1

There is a horizontal asymptote at y=1

4. Find the intercepts by setting x=0 and y=0

ƒ(0)=0/(0+1)=0

The graph passes through the origin (0,0)

5. Determine 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 like (−2,2)

Final Answer

ƒ(x)=x/(x+1)

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