Continuity and DifferentiabilityHard
Question
Let f : R → R be any function. Defion g : R → R by
Options
A.onto if f is onto
B.one-one if f is one-one
C.continuous if f is continuous
D.differentiable if f differentiable
Solution
Let h(x) = |x|, then
g(x) = |f(x)| = h{f(x)}
Since, composition of two continuous function is continuous, g is continuous if f is continuous.
So, answer is (c).
(a) is wrong answer. Let f(x) = x ⇒ g(x) =|x|
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 od g(x) is [0, ∞) but co-domain is given R.
(b) Let f(x) = x ⇒ g(x) = |x| Now, f(x) is one-one function but g(x) is many-one function. Hence, (b) is wrong.
(d) Let f(x) = x ⇒ g(x) = |x| Now, f(x) is differentiable for all x ∈ R but g(x) = |x| is not differentiable at x = 0 Hence, (d) is wrong.
g(x) = |f(x)| = h{f(x)}
Since, composition of two continuous function is continuous, g is continuous if f is continuous.
So, answer is (c).
(a) is wrong answer. Let f(x) = x ⇒ g(x) =|x|
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 od g(x) is [0, ∞) but co-domain is given R.
(b) Let f(x) = x ⇒ g(x) = |x| Now, f(x) is one-one function but g(x) is many-one function. Hence, (b) is wrong.
(d) Let f(x) = x ⇒ g(x) = |x| Now, f(x) is differentiable for all x ∈ R but g(x) = |x| is not differentiable at x = 0 Hence, (d) is wrong.
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