Let's prove that f is increasing on [-1,1] hence there are no a and b (a!=b) such that f(a)=f(b)
On [-1,0], x<0 and ln(|x|+1)>0
For any a<b<0
Multiply the inequalities with positive terms.
and f is increasing on [-1,0]
x is a positive increasing function.
|x| is an increasing function therefore 1+|x| is an increasing function and 1+|x|>1
Hence ln(1+|x|) is a positive increasing function. The product of 2 positive increasing functions is an increasing function.
f is an increasing function on (0,1]
In conclusion, f is increasing on [-1,1] therefore there are no `a!=b` such that f(a)=f(b).