Find the Domain

Problem

(9*w2−49)/(3*w2−w−14)⋅(7*w2+6*w−16)/(49*w2−64)

Solution

1. Identify the condition for the domain of a rational expression. The domain consists of all real numbers except those that make any denominator in the expression equal to zero.

2. Factor the first denominator 3*w2−w−14 We look for two numbers that multiply to −42 and add to −1 These are −7 and 6

3*w2−7*w+6*w−14=0

w*(3*w−7)+2*(3*w−7)=0

(w+2)*(3*w−7)=0

3. Solve for the excluded values from the first denominator.

w+2=0⇒w=−2

3*w−7=0⇒w=7/3

4. Factor the second denominator 49*w2−64 using the difference of squares formula a2−b2=(a−b)*(a+b)

(7*w)2−8=0

(7*w−8)*(7*w+8)=0

5. Solve for the excluded values from the second denominator.

7*w−8=0⇒w=8/7

7*w+8=0⇒w=−8/7

6. Combine all excluded values to state the domain. The domain is all real numbers w such that w≠−2,7/3,8/7,−8/7

Final Answer

Domain={[w∈ℝ,w≠−2,7/3,8/7,−8/7]}

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