Find the domain and range of the function f(x)=(x2+2x+1)/(x2-8x+12)

1.	Find the domain and range of the function f(x)=(x2+2x+1)/(x2-8x+12)
Answer» Ans. f(x) = (x2+2x+1)/(x2-8x+12)To find domain as the function is rational so its denominator must not be equal to zero(0).x2-8x+12\xa0= 0=> x2-6x-2x+12=0=> x(x-6)-2(x-6)=0=> (x-6)(x-2)=0=> x = 2,6For these two values this function \'ll become undefined. So domain = R - {2,6}To Find Range:Let f(x)=y=> (x2+2x+1)/(x2-8x+12) = y=> x2+2x+1 = yx2+-8xy+12y=> yx2-8xy +12y -x2-2x-1\xa0=0\xa0=> (y-1)x2\xa0-(8y+2)x\xa0+(12y-1) = 0Now D\xa0≥ 0=> b2\xa0-4ac\xa0≥\xa00=> (8y+2)2\xa0- 4(y-1)(12y-1)\xa0≥ 0\xa0= (8y+2)2\xa0≥\xa04(12y2\xa0-13y +1)=> 64y2\xa0+ 4 + 32y\xa0≥ 48y2\xa0-52y +4=> 16y2\xa0+ 84y\xa0≥ 0divide by 4=> 4y2\xa0+ 21y\xa0≥ 0=> y(4y+21)\xa0≥ 0=> y\xa0≥ 0 and\xa04y +21\xa0≥ 0y\xa0≥ 0 and y\xa0≥ -21/4So domain = R - (-21/4, 0)

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