Question

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

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

Answer 1

The correct option is D one-one but not onto

$f\left(g\right(x\left)\right)=2g\left(x\right)-1$
$=2\left(\frac{x-\frac{1}{2}}{x-1}\right)-1=\frac{x}{x-1}$
$f\left(g\right(x\left)\right)=1+\frac{1}{x-1}$

$\therefore f\left(g\right(x\left)\right)$ is one-one, into.