Question

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

• A. onto if $f$ is into
• B. one-one if $f$ is one-one
• C. continuous if $f$ is continuous
• D. differentiable if $f$ is differentiable

Answer 1

The correct option is C

continuous if $f$ is continuous

Determine the relationship between $g\text{and}f$.

We have, $g\left(x\right)=\left|f\left(x\right)\right|$

This means that $g\left(x\right)$ will always have a positive value. It will always lie in the $1^{st}\text{or}{2}^{nd}$ quadrant.

A function is said to be continuous if the graph can be drawn without lifting the pen, off the paper i.e it should have no breaks.

Now, if $f\left(x\right)$ is continuous, then its modulus will also be continuous.

Therefore $g\left(x\right)$ will also be continuous, so Option (C) holds true.

Hence, the correct answer is Option (C).