Show that the Signum function f - R - R- given by f x -left begin matrix 1- - if - X -0 10- - i f - X -0 1-1- - if - X -0 llendmatrix -right- is neither one-one nor onto-

F:R → R,f(x)={ 1, if x gt; 0 0, if x =0 1, if x lt; 0 . It is seen that f(1)=f(2)=1 but 1 ≠ 2. Therefore, f is not one one. Now, as f(x) takes only thr ...

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Standard XII

Mathematics

Scalar Multiplication of a Matrix

Show that the...

Question

Show that the Signum function $f : R \to R,$ given by $f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 1, i f x > 0 0, i f x = 0 - 1, i f x < 0 \end{matrix}$ is neither one-one nor onto.

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Solution

$f : R \to R, f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 1, i f x > 0 0, i f x = 0 - 1, i f x < 0 \end{matrix}$
It is seen that f(1)=f(2)=1 but $1 \neq 2$ . Therefore, f is not one-one.
Now, as f(x) takes only three values (1,0 or -1)for the element -2 in co-domain R, there does not exist any x in domain R such that f(x)=-2.
Therefore, f is not onto.
Hence, the Signum function is neither one-one nor onto.

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Show that the Signum function f:R→R, given by f(x)=⎧⎪⎨⎪⎩1, if x>00, if x=0−1, if x<0 is neither one-one nor onto.

Show that the Signum function $f : R \to R,$ given by $f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 1, i f x > 0 0, i f x = 0 - 1, i f x < 0 \end{matrix}$ is neither one-one nor onto.