Question

Let a function f be defined by f(x) = 1 2 x  x , x Є R Find (i) f ( x 1 ) x 0 (ii) f (2x) (iii) f (x – 1)

Answer 1

Answer:

Step-by-step explanation:

We have,` f(x)=(x)/(1+x^(2)), x in R`

Ist Method f(x) is an odd function and maximum occur at x = 1

From the graph it is clear that range of f(x) is

`[-(1)/(2),(1)/(2)]`

IInd Method `f(x)=(1)/(x+(1)/(x))`

If `x gt 0`, then by `AM ge GM`, we get `x+(1)/(x) ge 2`

`rArr (1)/(x+(1)/(x)) le (1)/(2) rArr 0 lt f(x) le (1)/(2)`

If `x lt 0`, then by `AM ge GM`, we get `x+(1)/(x) le -2`

`rArr (1)/(x+(1)/(x)) ge -(1)/(2) rArr -(1)/(2) le f(x) le 0`

If x = 0, then `f(x)=(0)/(1+0)=0`

Thus, `-(1)/(2) le f(x) le (1)/(2)`

Hence, `f(x) in [-(1)/(2),(1)/(2)]`

IIIrd Method

Let `y=(x)/(1+x^(2)) rArr yx^(2)-x+y=0`

` because x in R, " so "D ge 0`

`rArr 1-4y^(2) ge 0`

`rArr (1-2y)(1+2y) ge 0 rArr y in [-(1)/(2),(1)/(2)]`

So, range is `[-(1)/(2),(1)/(2)]`.

Answer 2

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