FunctionHard
Question
Let f : R → R be a function defined by f(x) = 
, then f is:
, then f is: Options
A.one - one but not onto
B.onto but not one - one
C.onto as well as one - one
D.neither onto nor one - one
Solution
One One / Many One
f(x) =
, Domain x ∧ R
f′(x) =
f′(x) =
> 0 ⇒ x ∈ (- ∞, 0) ∪ 
f′(x) < 0 ⇒ x ∈
f′(x) = 0 ⇒ x = 0,
Function is increasing and decreasing in different intervals, so non monotonic
∴ Many one function.
Onto / Into
f(x) =
2x2 - x + 5 > 0, ∀ x ∈ R and 7x2 + 2x + 10 > 0 ∀ x ∈ R
∵ a = 2 > 0 and ∵ a = 7 and D = 4 - 280 < 0
D = 1 - 40 = - 39 < 0
∴ f(x) > 0 ∀ x ∈ R
Also f(x) never tends to ± ∞as 7x2 + 2x + 10 has no real roots, Range ≠ Codomain so into function.
f(x) =
, Domain x ∧ R f′(x) =
f′(x) =
> 0 ⇒ x ∈ (- ∞, 0) ∪ f′(x) < 0 ⇒ x ∈
f′(x) = 0 ⇒ x = 0,
Function is increasing and decreasing in different intervals, so non monotonic
∴ Many one function.
Onto / Into
f(x) =
2x2 - x + 5 > 0, ∀ x ∈ R and 7x2 + 2x + 10 > 0 ∀ x ∈ R
∵ a = 2 > 0 and ∵ a = 7 and D = 4 - 280 < 0
D = 1 - 40 = - 39 < 0
∴ f(x) > 0 ∀ x ∈ R
Also f(x) never tends to ± ∞as 7x2 + 2x + 10 has no real roots, Range ≠ Codomain so into function.
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