Question

Let $f:R^{+}\to \{-1,0,1\}$ defined by $f\left(x\right)=sgn(x-x^4+x^7-x^8-1)$ where sgn denotes signum function. Then $f\left(x\right)$ is

• A. many one and onto
• B. many-one and into
• C. one-one and onto
• D. one-one and into

Answer 1

The correct option is B

many-one and into

$f\left(x\right)=sgn(x-x^4+x^7-x^8-1)$

for $xϵ(0,1);x-1<0,x^7-x^4<0$

$\therefore x-x^4+x^7-x^8-1<0$

For $xϵ(1,\infty );x<x^4,x^7<x^8$

$\therefore x-x^4+x^7-x^8-1<0$

Also for $x=1,x-x^4+x^7-x^8-1=-1$

Thus, $x-x^4+x^7-x^8-1<0$ for all $xϵR^{+}$

Hence, $sgn(x-x^4+x^7-x^8-1)=-1\forall xϵR^{+}$

Therefore, f(x) is many-one and into functions.