1. Tardigrade
  2. Question
  3. Mathematics
  4. Let f: R+ arrow -1,0,1 defined by f(x)=sgn(x-x4+x7-x8-1) where sgn denotes signum function. Then f(x) is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

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

Relations and Functions - Part 2

Solution:

$f(x)=sgn\left(x-x^{4}+x^{7}-x^{8}-1\right)$
For $x \in(0,1) ; x-1 < 0, x^{7}-x^{4} < 0$
$\therefore x-x^{4}+x^{7}-x^{8}-1 < 0$
For $x \in(1, \infty) ; x < x^{4}, x^{7} < x^{8}$
$\therefore x-x^{4}+x^{7}-x^{8}-1 < 0$
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 $x \in R^{+}$
Hence sgn$\left(x-x^{4}+x^{7}-x^{8}-1\right)=-1 \forall x \in R^{+}$.
Therefore $f(x)$ is many-one and into.