Graph x/(x-3)

Problem

ƒ(x)=x/(x−3)

Solution

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

x−3=0⇒x=3

The domain is all real numbers except x=3

2. Find the vertical asymptote by setting the denominator to zero.
   There is a vertical asymptote at x=3

3. Find the horizontal asymptote by comparing the degrees of the numerator and denominator.
   Since the degrees are equal, the asymptote is the ratio of the leading coefficients.

y=1/1=1

4. Find the x-intercept by setting the numerator to zero.

x=0

The x-intercept is (0,0)

5. Find the y-intercept by evaluating the function at x=0

ƒ(0)=0/(0−3)=0

The y-intercept is (0,0)

6. Determine the behavior of the graph by testing points on either side of the vertical asymptote.
   For x<3 let x=2

ƒ(2)=2/(2−3)=−2

For x>3 let x=4

ƒ(4)=4/(4−3)=4

Final Answer

ƒ(x)=x/(x−3)

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