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 onto
B.
one-one if $$f$$ is one-one
C.
continuous if $$f$$ is continuous
D.
differentiable if $$f$$ is differentiable.
Answer :
continuous if $$f$$ is continuous
Solution :
$$\eqalign{
& {\text{Let}}\,h\left( x \right) = \left| x \right|\,{\text{then}} \cr
& g\left( x \right) = \left| {f\left( x \right)} \right| = h\left( {f\left( x \right)} \right) \cr} $$
Since composition of two continuous functions is continuous, therefore $$g$$ is continuous if $$f$$ is continuous.