Find the Domain and Range f(x)=(x^2+9x-9)/(x-9)

Problem

ƒ(x)=(x2+9*x−9)/(x−9)

Solution

1. Identify the domain by finding values of x that make the denominator zero.

2. Set the denominator equal to zero: x−9=0

3. Solve for x to find the excluded value: x=9

4. State the domain as all real numbers except x=9

5. Analyze the range by checking for horizontal asymptotes or holes. Since the degree of the numerator is exactly one higher than the degree of the denominator, there is an oblique asymptote and the function covers all real values except for a possible gap.

6. Rewrite the function using polynomial long division: ƒ(x)=x+18+153/(x−9)

7. Find the local extrema to determine the range. Set the derivative d(ƒ(x))/d(x)=1−153/((x−9)2) equal to zero.

8. Solve for x: (x−9)2=153 which gives x=9±√(,153)=9±3√(,17)

9. Evaluate the function at these points to find the relative maximum and minimum: ƒ*(9−3√(,17))=27−6√(,17) and ƒ*(9+3√(,17))=27+6√(,17)

10. State the range as the set of values excluding the interval between these two local extrema.

Final Answer

Domain: *{[x∈ℝ,x≠9]}

Range: *{[y∈ℝ,y≤27−6√(,17)* or *y≥27+6√(,17)]}

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