Question

Let $f: R \rightarrow R$ be defined as $f(x)=2 x-1$ and $g: R-\{1\} \rightarrow R$ be defined as $g(x)=\frac{x-\frac{1}{2}}{x-1}$ Then the composition function $f(g(x))$ is :

• A. onto but not one-one
• B. both one-one and onto
• C. one-one but not onto
• D. neither one-one nor onto

Answer 1

Answer: (C)

$f(g(x))=2 g(x)-1=2\left(\frac{2 x-1}{2(x-1)}\right)-1 $
$=\frac{x}{x-1}=1+\frac{1}{x-1} $
Range of $f(g(x)= R -\{1\}$
Range of $f(g(x))$ is not onto
$\&$ $f( g ( x ))$ is one-one
So $f(g(x))$ is one-one but not onto.