FunctionHard
Question
Let function f : R → R be defined by f(x) = 2x + sin x for x ∈ R Then, f is
Options
A.one-to-one and onto
B.one-to-one but not onto
C.onto but not one-to-one
D.neither one-to-one nor onto
Solution
Given, f(x) = 2x + sin x
f′(x) = 2 + cos x ⇒ f′(x) > 0, ∀ x ∈ R
Which shows f(x) is one-one, as f(x) is strictly increasing.
Since, f (x) is increasing for every x ∈ R
∴ f(x) takes all intermediate values between (-∞, ∞)
∴ Range of f(x) ∈ R
f′(x) = 2 + cos x ⇒ f′(x) > 0, ∀ x ∈ R
Which shows f(x) is one-one, as f(x) is strictly increasing.
Since, f (x) is increasing for every x ∈ R
∴ f(x) takes all intermediate values between (-∞, ∞)
∴ Range of f(x) ∈ R
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