Find the domain and range of …
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Posted by Sourav Keshri 8 years, 10 months ago
- 1 answers
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Naveen Sharma 8 years, 10 months ago
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 = 0
=> x2-6x-2x+12=0
=> x(x-6)-2(x-6)=0
=> (x-6)(x-2)=0
=> x = 2,6
For 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 =0
=> (y-1)x2 -(8y+2)x +(12y-1) = 0
Now D ≥ 0
=> b2 -4ac ≥ 0
=> (8y+2)2 - 4(y-1)(12y-1) ≥ 0
= (8y+2)2 ≥ 4(12y2 -13y +1)
=> 64y2 + 4 + 32y ≥ 48y2 -52y +4
=> 16y2 + 84y ≥ 0
divide by 4
=> 4y2 + 21y ≥ 0
=> y(4y+21) ≥ 0
=> y ≥ 0 and 4y +21 ≥ 0
y ≥ 0 and y ≥ -21/4
So domain = R - (-21/4, 0)
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