Question

Find all possible values of a function f(x)=1-x^2/x^2+3​

Answer 1

Answer:

1−f(x)−f(x)  

2

>f(1−5x)

Or  

f(f(x))>f(1−5x) ...(i)

Now  

f  

′

(x)=−1−3x  

2

 

Hence

f  

′

(x)<0 for all x. ...(ii)

Hence

f(f(x)>f(1−5x)

But f(x) is a decreasing function (from ii).

Hence

f(x)<1−5x

Or  

1−x−x  

3

<1−5x.

Or  

4x−x  

3

<0

Or  

x  

3

−4x>0

Or  

x(x  

2

−4)>0

Or  

x>0 and x  

2

−4>0 or  

x<0 and x  

2

−4<0

Hence if x>0 x  

2

−4>0 implies xϵ(2,∞)

And  

x<0 and x  

2

−4<0 implies (−2,0)

Hence

xϵ(−2,0)∪(2,∞).