Mapping f:R R which is defined as f(x) = x, ;x R will be - Vidyadip

MCQ

Mapping $f:R \to R$ which is defined as $f(x) = \cos x,\;x \in R$ will be

✓
Neither one-one nor onto
B
One-one
C
Onto
D
One-one onto

✓

Answer

Correct option: A.

Neither one-one nor onto

a
(a) Let ${x_1},\,{x_2} \in R,$ then $f({x_1}) = \cos {x_1}$, $f({x_2}) = \cos {x_2}$,

so $f({x_1}) = f({x_2})$

$⇒$ $\cos {x_1} = \cos {x_2}$ $⇒$ ${x_1} = 2n\pi \pm {x_2}$

$⇒$ ${x_1} \ne {x_2}$, so it is not one-one.

Again the value of $f$-image of $x$ lies in between $-1$ to $1$

$⇒$ $f[R] = \left\{ {f(x): - 1 \le f(x) \le 1)} \right\}$

So other numbers of co-domain (besides $-1$ and $1$) is not $f$ -image.

$f[R] \in R,$ so it is also not onto.

So this mapping is neither one-one nor onto.

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