Let $f : R \to R$ be a function. Define $g : R \to R$ by $g (x) = | f (x) |$ for all $x$ . Then, $g$ is:

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Continuous if $f$ is continuous

Differentiable if $f$ is differentiable

Onto it $f$ is onto

One-one if $f$ is one-one

answer is C.

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Detailed Solution

$g (x) = | f (x) | \geq 0$ . So, $g (x)$ cannot be onto. If $f (x)$ is one-one and $f (x_{1}) = - f (x_{2})$ then, $g (x_{1}) = g (x_{2})$ . So, $f (x)$ is non-one’ does not ensure that $g (x)$ is one-one.

Question Image

If $f (x)$ is continuous for $x \in R$ , $| f (x) |$ is also continuous for $x \in R$ . This is obvious from the following graphical consideration.

So the answer (c) is correct. The fourth answer (d) is not correct from the above graphs $y = f (x)$ is differentiable at $P$ while $y = | f (x) |$ has two tangents at $P$ , i.e., not differentiable at $P$ .