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Show that f : [–1, 1] → R, given by f (x) = x/(x+2) is one-one. Find the inverse of the function f : [–1, 1] → Range f. some x in [–1, 1], i.e., x = 2y/(1-y).
Show that f : [–1, 1] → R, given by f (x) = x/(x+2) is one-one. Find the inverse of the function f : [–1, 1] → Range f. some x in [–1, 1], i.e., x = 2y/(1-y).
CBSE | Class 12 | Exercise 1.3 | Maths | NCERT | Relations and Functions
Show that f : [–1, 1] → R, given by f (x) = x/(x+2) is one-one. Find the inverse of the function f : [–1, 1] → Range f. some x in [–1, 1], i.e., x = 2y/(1-y).

solution:

Given capacity: (x) = x/(x+2) Let x, y ∈ [–1, 1]

Let f(x) = f(y)

x/(x+2) = y/(y+2) xy + 2x = xy + 2y x = y

f is one-one. Once more,

Since f : [–1, 1] → Range f is onto say, y = x/(x+2)

yx + 2y = x x(1 – y) = 2y or x = 2y/(1-y)

x = f – 1 (y) = 2y/(1-y); y not equivalent to 1

f is onto capacity, and f – 1(x) = 2x/(1-x).

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