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

Problem

ƒ(x)=(x2)/(x2−4)

Solution

1. Identify the domain by finding where the denominator is not equal to zero.

x2−4≠0

2. Solve for x to find the excluded values.

(x−2)*(x+2)≠0

x≠2,x≠−2

3. Express the domain in interval notation.

Domain:(−∞,−2)∪(−2,2)∪(2,∞)

4. Identify the range by setting y=ƒ(x) and solving for x in terms of y

y=(x2)/(x2−4)

y*(x2−4)=x2

y*x2−4*y=x2

y*x2−x2=4*y

x2*(y−1)=4*y

x2=(4*y)/(y−1)

5. Determine restrictions on y for x to be a real number.

(4*y)/(y−1)≥0

6. Solve the inequality using critical points y=0 and y=1

Intervals:(−∞,0],(0,1),[1,∞)

Testing values shows the expression is non-negative when y≤0 or y>1 Note that y≠1 because it would make the denominator zero.

7. Express the range in interval notation.

Range:(−∞,0]∪(1,∞)

Final Answer

Domain:(−∞,−2)∪(−2,2)∪(2,∞),Range:(−∞,0]∪(1,∞)

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