# SOLUTION: What is the domain and range of the following rational expression? f(x)=x+3 _____ x^2-5x-66

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 Question 89316: What is the domain and range of the following rational expression? f(x)=x+3 _____ x^2-5x-66Answer by Edwin McCravy(8906)   (Show Source): You can put this solution on YOUR website!``` What is the domain and range of the following rational expression? f(x)=x+3 _____ x^2-5x-66 f(x) = Factor the denominator: f(x) = The domain contains all numbers which can be substituted for x, and an answer is defined. Denominators cannot be 0 so x cannot be any number that will cause the denominator to be 0. DENOMINATOR = (x-11)(x+2) So x cannot equal to 11 or -2 So the domain is or in interval notation (-,-2)U(-2,11)U(11,) Now we have to find the range. To do that we first replace f(x) by y and solve for x: f(x) = y = Multiply both sides by the LCD = = = Get 0 on the right: = Get the x terms together: = Factor x out of the second and third terms = factor -3 out of the last two terms on the left: = This is a quadratic equation in x with a = y, b = -(5y+1), c = -3(22y+1) We could solve for x, but we only need the discriminant b²-4ac to be non-negative, so b²-4ac = [-(5y+1)]² - 4(y)[-3(22y+1)] = 25y² + 10y + 1 + 12y(22y+1) = 25y² + 10y + 1 + 264y² + 12y = 289y² + 22y + 1 > = 0 The left side has discriminant 22²-4(289)(1) = -672 thus it has no real zeros and represents a parabola which is always positive. Therefore the range is "all real numbers" or (-,) Edwin```