Let f : ℝ → ℝ be a function such that:
|f(x) − f(y)| ≤ ½ |x − y|, for all x, y ∈ ℝ
f′(x) ≥ ½, for all x ∈ ℝ, and f(1) = ½
Find the number of points of intersection of the curves:
y = f(x) and y = x² − 2x − 5
From the given:
So assume:
f(x) = (1/2)x + c
Use the point: f(1) = 1/2 ⇒ (1/2)·1 + c = 1/2 ⇒ c = 0
Therefore:
f(x) = (1/2)x
Solve:
(1/2)x = x² − 2x − 5
⇒ x² − (5/2)x − 5 = 0
⇒ 2x² − 5x − 10 = 0
⇒ x = [5 ± √(25 + 80)] / 4
⇒ x = [5 ± √105] / 4
Therefore, there are 2 real points of intersection.
Let A be the set of 30 students of class XII in a school. Let f : A -> N, N is a set of natural numbers such that function f(x) = Roll Number of student x.
On the basis of the given information, answer the followingIs \( f \) a bijective function?