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Let \; f:R\rightarrow R\; be defined as f(x )=2x -1 and g:R-{1}\rightarrow R\; be defined as g(x)=\frac{x-\frac{1}{2}}{x-1}.Then the composition function f\left ( g(x) \right ) is:
Option: 1 one-one but not onto
Option: 2 neither one-one nor onto
Option: 3 onto but not one-one
Option: 4 both one-one and onto

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\begin{aligned} &f(g(x))=2 g(x)-1\\ &=2 \frac{\left(x-\frac{1}{2}\right)}{x-1}=\frac{x}{x-1}\\ &f(g(x))=1+\frac{1}{x-1}\\ &\text { one-one, into } \end{aligned}

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