Find the Domain and Range y=x/(x^2-4)

Problem

y=x/(x^2−4)

Solution

1. Identify the condition for the domain by setting the denominator not equal to zero, as division by zero is undefined.

x^2−4≠0

2. Solve for x by factoring the difference of squares.

(x−2)*(x+2)≠0

x≠2,x≠−2

3. State the domain in interval notation based on the excluded values.

Domain:(−∞,−2)∪(−2,2)∪(2,∞)

4. Analyze the range by examining the behavior of the function. Since the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at y=0

5. Determine if the function crosses the horizontal asymptote by setting y=0

0=x/(x^2−4)

x=0

6. Conclude that because the function is continuous on its defined intervals and passes through y=0 and because the vertical asymptotes at x=2 and x=−2 cause the function to approach ±∞ the range covers all real numbers.

Range:(−∞,∞)

Final Answer

Domain:(−∞,−2)∪(−2,2)∪(2,∞),Range:(−∞,∞)

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