Question

Let f : R → R be and function. Define g : R → R 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) differentiable if f is differentiable

Answer 1

Correct option (c) continuous if f is continuous

Explanation :

Let h(x) = | x |, then

Since, composition of two continuous functions is continuous , g is continuous if f is continuous. So, answer is (c).

(a) is wrong answer.

Now, f(x) is an onto function. Since, co-domain of x is R and range of x is R. But g(x) is into function. Since, range of g(x) is [0, ∞) but co-domain is given R.

Now, f(x) is one-one function but g(x) is many-one function. Hence, (b) is wrong.

Now, f(x) is differentiable for all x ∈ R but g(x) = |x| is not differentiable at x = 0 Hence, (d) is wrong.