Question

Let $g\left(x\right)=1+x-\left[x\right]andf\left(x\right)=\{\begin{array}{cc}-1,& x<0\\ 0,& x=0\\ 1,& x>0\end{array}$
Then for all x, f(g(x)) is equal to

• A. x
• B. 1
• C. f(x)
• D. g(x)

Answer 1

The correct option is B 1

$g\left(x\right)=1+x-\left[x\right];$
$f\left(x\right)=\{\begin{array}{cc}-1,& x<0\\ 0,& x=0\\ 1,& x>0\end{array}$
For integral values of x; g(x) = 1
For x < 0; (but not integral value) $x-\left[x\right]>0\Rightarrow g\left(x\right)>1$
For x > 0; (but not integral value) $x-\left[x\right]>0\Rightarrow g\left(x\right)>1$
$\therefore g\left(x\right)\ge 1,\forall x$ $\therefore f\left(g\right(x\left)\right)=1,\forall x$