Find the Domain f(x)=(2x^2-4x+2)/(3x^2-3x-6)

Problem

ƒ(x)=(2*x2−4*x+2)/(3*x2−3*x−6)

Solution

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

3*x2−3*x−6≠0

2. Factor out the greatest common factor from the quadratic expression in the denominator to simplify the equation.

3*(x2−x−2)≠0

3. Factor the quadratic trinomial inside the parentheses by finding two numbers that multiply to −2 and add to −1

3*(x−2)*(x+1)≠0

4. Solve for the excluded values of x by setting each factor containing a variable to zero.

x−2=0⇒x=2

x+1=0⇒x=−1

5. State the domain by excluding these values from the set of all real numbers.

Domain={x|x∈ℝ,x≠−1,x≠2}

Final Answer

Domain=(−∞,−1)∪(−1,2)∪(2,∞)

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