Let us define a function y = f(x): X → Y. If we define a function g(y) such that x = g(y) then g is said to be the inverse function of 'f'.
Think: If f is many-to-one, g : Y → X will not satisfy the definition of a function.
So to define the inverse of a function, it must be one-one.
Further if f : X → Y is into then there must be a point in Y for which there is no x. This again violates the definition of function for 'g' (In fact when f is one tone and onto then 'g' can be defined from range of f to domain of i.e. g : f(X) → X.
Hence, the inverse of a function can be defined within the same sets for x and Y only when it is one-one and onto or Bijective.
Note: A monotonic function i.e. bijection function is always invertible.
Illustration: Let f : R → R be defined as
y = f(x) = x2. Is it invertible?
No it is not invertible because this is a many one into function
This is many-one because for x = + a, y = a2, this is into because y does not take the negative real values.
Illustration: Let f : R → [0, α) be defined as y = f(x) = x2. Is it invertible?
(see figure below)
No it is not invertible, it because it is many one onto function.
Illustration: Let f : [0, α) → [0, α) be defined as y = f(x) = x2. Is it invertible? If so find its inverse.
Yes, it is invertible because this is bijection function. Its graph is shown in figure given below.
Let y = x2 (say f(x))
=> x = +√y
But x is positive, as domain of f is [0, α)
=> x = + √y
Therefore Inverse is y = √x = g(x)
f(g(x)) = f(√x) = x, x> 0
g(f(x)) = g(x2) = √x2 = x, x > 0
i.e. if f and g are inverse of each other then f(g(x)) = g(f(x)) = x
Illustration: How are the graphs of function and the inverse function related? These graphs are mirror images of each other about the line y = x.
Also, if the graph of y = f(x) and y = f-1 (x), they intersect at the point where y meet the line y = x.
Graphs of the function and its inverse are shown in figures given above as Figure (A) and (B)
For Figure (A)
y = f(x) = x2; f : [0, ∞) → [0, ∞)
1. Define y = f(x) = x2 in some other ay so that its inverse is possible.
2. What is the inverse of y = loge (x + √(x2 + 1))
Ans.1 f : (-α, 0] → [0, α)
y = f(x) = x2 and its inverse is
y = -√x (Figure B)
Ans.2 y = (ex - ex)/2
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