What is the domain and range of y = 1/(x-1)(x+2)?

AI Thread Summary
The function y = 1/((x-1)(x+2)) has a domain of all real numbers except x = -2 and x = 1. The range is y ≤ -4/9 or y > 0. A method to find the range involves analyzing the discriminant of the equation y(x^2 + x - 2) = 1, leading to the conclusion about y's values. Suggestions for alternative methods include using graphing or making a substitution to simplify the function. Overall, the discussion emphasizes finding different approaches to determine the domain and range of the function.
sinjan.j
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Homework Statement



y = \frac{1}{(x-1)(x+2)}

find out the domain and range

ans: y<= -4/9 or y>0

x~E~R~-\{-2,1\}





The Attempt at a Solution



I know the solution and I used this methoD:

y(x^2 + x -2) =1 (where y\neq0 )

when x is real, the discriminant

y^2 -4y(-2y-1) >=0 (where y\neq0 )

9y^2 + 4y >= 0 (where y\neq0 )

y(9y +4) >= 0 (where y\neq0 )

y<= -4/9 or y>0


My request is ... please use some other easy method to illustrate this problem.. I want to know other methods of solving this same problem.. please help.

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This is the simplest way that I know of. If graphing is simple enough, you can try.
 
Welcome to PF!

sinjan.j said:

Homework Statement



y = \frac{1}{(x-1)(x+2)}

find out the domain and range

Hi sinjan.j! Welcome to PF! :smile:

Easy method:

It looks nearly symmetric …

so adjust it to make it symmetric! :smile:

Hint: the range will be the same even if you make a substitution for x. Try w = x - a, where you choose a to make 1/(x-1)(x+2) symmetric in w.
 
simpler method = calculator
 
Thanx everbody

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