Graph f(x)=1/x

Problem

ƒ(x)=1/x

Solution

1. Identify the domain by finding values where the denominator is zero. Since x≠0 the domain is (−∞,0)∪(0,∞)

2. Determine the asymptotes of the function. There is a vertical asymptote at x=0 because the function is undefined there, and a horizontal asymptote at y=0 because (lim_x→±∞)(1/x)=0

3. Analyze the behavior in different quadrants. For x>0 ƒ(x) is positive and decreasing, occupying the first quadrant. For x<0 ƒ(x) is negative and increasing, occupying the third quadrant.

4. Plot key points to define the shape of the hyperbola. For example, plot (1,1) (2,0.5) (0.5,2) (−1,−1) (−2,−0.5) and (−0.5,−2)

5. Sketch the curves approaching the asymptotes. Draw two separate smooth branches that never touch the axes.

Final Answer

ƒ(x)=1/x

The graph is a rectangular hyperbola with a vertical asymptote at x=0 and a horizontal asymptote at y=0

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