Find the Domain

Problem

ƒ(x)=(x2−4)/(x2+x−6)

Solution

1. Identify the condition for the domain of a rational function, which is that the denominator cannot be equal to zero.

x2+x−6≠0

2. Factor the quadratic expression in the denominator to find the values of x that must be excluded.

(x+3)*(x−2)≠0

3. Solve for the excluded values by setting each factor to zero.

x+3=0⇒x=−3

x−2=0⇒x=2

4. State the domain as all real numbers except for the values that make the denominator zero, regardless of whether factors cancel out in the simplified expression.

Domain={[x∈ℝ,x≠−3,x≠2]}

Final Answer

Domain=(−∞,−3)∪(−3,2)∪(2,∞)

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