Graph f(x)=(x^2-4)/(x^2-9)

Problem

ƒ(x)=(x2−4)/(x2−9)

Solution

1. Find the domain by identifying where the denominator is zero.

x2−9=0

x=±3

The domain is (−∞,−3)∪(−3,3)∪(3,∞)

2. Identify vertical asymptotes by setting the denominator to zero (since the numerator is not zero at these points).

x=3

x=−3

3. Identify the horizontal asymptote by comparing the degrees of the numerator and denominator.

(lim_x→∞)((x2−4)/(x2−9))=1

y=1

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

x2−4=0

x=±2

The intercepts are (2,0) and (−2,0)

5. Find the y-intercept by evaluating ƒ(0)

ƒ(0)=(0−4)/(0−9)

ƒ(0)=4/9

The intercept is (0,4/9)

6. Determine symmetry by checking if the function is even or odd.

ƒ*(−x)=((−x)2−4)/((−x)2−9)=(x2−4)/(x2−9)

Since ƒ(x)=ƒ*(−x) the graph is symmetric about the y-axis.

7. Analyze behavior between asymptotes.
   For x∈(−3,3) the function is below the horizontal asymptote y=1 because the denominator is negative and the numerator is between −4 and 5
   For x∈(−∞,−3)∪(3,∞) the function is above the horizontal asymptote y=1

Final Answer

ƒ(x)=(x2−4)/(x2−9)

The graph has vertical asymptotes at x=±3 a horizontal asymptote at y=1 x-intercepts at (±2,0) and a y-intercept at (0,4/9)

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