Mathematics INVERSE OF A FUNCTION

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INVERSE OF A FUNCTION

Inverse of a Function

Let `f: A-> B` be a one- one & onto function, then their exists a unique function

`g: B-> A` such that `f(x) = y Leftrightarrow g(y) = x, AA x in A` & `y in B`. Then `g` is said to be inverse off.

Thus `g = f^-1: B-> A = {(f(x), x) I (x, f(x)) in f}`.

Consider a one-one onto function with domain `A= {a, b, c}` & range `B = {I , 2, 3}`

Domain of `f= {a, b, c} =` Range of `f^-1`

Range of `f = {1 , 2, 3} =` Domain of `f^- 1`

Note:

(a) Only one-one onto functions (i.e., Bijections) are invertible.

(b) To find the inverse

Step-1 : write `y = f(x)`

Step-2: solve this equation for `x` in terms of `y` (if possible)

Step-3: To express `f^-1` as a function of `x`, interchange `x` and `y`.

Let `f: A-> B` be a one- one & onto function, then their exists a unique function

`g: B-> A` such that `f(x) = y Leftrightarrow g(y) = x, AA x in A` & `y in B`. Then `g` is said to be inverse off.

Thus `g = f^-1: B-> A = {(f(x), x) I (x, f(x)) in f}`.

Consider a one-one onto function with domain `A= {a, b, c}` & range `B = {I , 2, 3}`

Domain of `f= {a, b, c} =` Range of `f^-1`

Range of `f = {1 , 2, 3} =` Domain of `f^- 1`

Note:

(a) Only one-one onto functions (i.e., Bijections) are invertible.

(b) To find the inverse

Step-1 : write `y = f(x)`

Step-2: solve this equation for `x` in terms of `y` (if possible)

Step-3: To express `f^-1` as a function of `x`, interchange `x` and `y`.

Properties of inverse of a function

(i) The inverse of Bijection is unique.

(ii) The inverse of Bijection is also bijection.

(iii) If `f: A-> B` is Bijection & `g: B->A` is inverse of `f` , then `fog = I_B` & `gof= I_A`, where `I_A, I_B` are the identical function on the set `A` and `B` respectively

(iv) If `f: A-> B` and `g: B-> C` are two bijections, then `gof: A-> C` is bijections and `(gof)^-1 = f^-1og^-1` .

(v) In general `fog != gof` but if `fog = gof` then either `f^-1 = g` or `g^-1 = f` also `(fog)(x) = (gof) (x) = x`.

(vi) The graphs of `f` & `g` are the mirror images of each other in the line `y = x`. As shown in the figure given below a point `(x ', y') ` corresponding to `y= x^2(x>= 0)` changes to `(y', x')` corresponding to `y = +sqrtx` , the changed forrn of `x = sqrty`.

(i) The inverse of Bijection is unique.

(ii) The inverse of Bijection is also bijection.

(iii) If `f: A-> B` is Bijection & `g: B->A` is inverse of `f` , then `fog = I_B` & `gof= I_A`, where `I_A, I_B` are the identical function on the set `A` and `B` respectively

(iv) If `f: A-> B` and `g: B-> C` are two bijections, then `gof: A-> C` is bijections and `(gof)^-1 = f^-1og^-1` .

(v) In general `fog != gof` but if `fog = gof` then either `f^-1 = g` or `g^-1 = f` also `(fog)(x) = (gof) (x) = x`.

(vi) The graphs of `f` & `g` are the mirror images of each other in the line `y = x`. As shown in the figure given below a point `(x ', y') ` corresponding to `y= x^2(x>= 0)` changes to `(y', x')` corresponding to `y = +sqrtx` , the changed forrn of `x = sqrty`.

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