Question

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

• A. Onto if f is onto
• B. One-one if f is one-one
• C. Continuous if f is continuous
• D. None of these

Answer 1

The correct option is C Continuous if f is continuous

$g\left(x\right)=|f\left(x\right)|\ge 0.Sog\left(x\right)$ cannot be onto. if f(x) one-one and $f\left(x_1\right)=-f\left(x_2\right)$ then
$g\left(x_1\right)=g\left(x_2\right)$.
So, 'f(x) is one-one' does not ensure that g(x) is one-one.

If f(x) is continuous for $x\in R,\left(x\right)$ is also continuous for $x\in R$.
This is obvious from the following graphical consideration.
So g is continuous if f is continuous.