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Question: f : R → R where then f(x) = \(\frac{x^{2} + 3x + 6}{x^{2} + x + 1}\) is...

f : R → R where then f(x) = x2+3x+6x2+x+1\frac{x^{2} + 3x + 6}{x^{2} + x + 1} is

A

Many one and into

B

Many one and on to

C

One one and into

D

One one and on-to

Answer

Many one and into

Explanation

Solution

Clearly domain of f(x) is all real numbers f(x)

= −2x2+10x−3(x2+x+1)2=−2(x2−10x+3)(x2+x+1)2\frac { - 2 x ^ { 2 } + 10 x - 3 } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } } = \frac { - 2 \left( x ^ { 2 } - 10 x + 3 \right) } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } }

Now since discriminant of 2x2 − 10x + 3 is positive thus f'(x) will not retain it's sign throughout the domain. Hence f(x) is many one. Also Lim⁡x→±∞\operatorname { Lim } _ { x \rightarrow \pm \infty } f(x) = 1 thus f(x) is into.