Let f:(0,∞)→R be defined as f(x)=tan−1(log2(x)) and g:R→R be defined as g(x)=2x2−3x+22x2+3x+2
If f∘g denotes the composition of f and g, then
f∘g(x)=tan−1(log2(2x2−3x+22x2+3x+2))
Replacing x by −1x, we get
f∘g(−1x)=tan−1(log2(2x2+3x+22x2−3x+2))
Replacing x by −x, we get
f∘g(−x)=tan−1(log2(2x2+3x+22x2−3x+2))
⇒f∘g(−1x)=f∘g(−x)
As we already proved that f∘g(−1x)=f∘g(−x), so it can't be strictly increasing in the given domain (−∞,∞).
So, f∘g(x) is many-one function.
Now, Let y=2x2−3x+22x2+3x+2
⇒(2y−2)x2+(3y+3)x+2y−2=0
Now D≥0
(3y+3)2−4(2y−2)(2y−2)≥0
⇒9y2+18y+9−16y2+32y−16≥0
⇒−7y2+50y−7≥0
⇒7y2−50y+7≤0
⇒y∈[17,7]