Lecture 3 : Composition of function and invertible function

:  

Letbe a function defined as. The inverse of fis map g: Range

(A)  (B) 

(C)  (D) 

If f: R Ã¢â€ ’ R be given by, then fof(x) is

(A)  (B) x³ (C) x (D) (3 - x³)

Let f: X → Y be an invertible function. Show that the inverse of f^-1 is f, i.e.,

(f^-1)^-1 = f.

Consider f: {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f^-1and show that (f^-1)^-1= f.

Let f: X → Y be an invertible function. Show that f has unique inverse.

(Hint: suppose g₁ and g₂ are two inverses of f. Then for all y ∈ Y,

fog₁(y) = I_Y(y) = fog₂(y). Use one-one ness of f).

Consider f: R₊→ [ - 5, ∞) given by f(x) = 9x²+ 6x - 5. Show that fis invertible with.

Consider f: R₊ Ã¢â€ ’ [4, ∞) given by f(x) = x²+ 4. Show that fis invertible with the inverse f ^{- 1}of given f by, where R₊is the set of all non-negative real numbers.

Consider f: R → Rgiven by f(x) = 4x + 3. Show that fis invertible. Find the inverse of f.

Show that f: [ - 1, 1] → R, given byis one-one. Find the inverse of the function f: [ - 1, 1] → Range f.

(Hint: For y ∈Range f, y =, for some x in [ - 1, 1], i.e.,)

State with reason whether following functions have inverse

(i) f: {1, 2, 3, 4} → {10} with

f = {(1, 10), (2, 10), (3, 10), (4, 10)}

(ii) g: {5, 6, 7, 8} → {1, 2, 3, 4} with

g = {(5, 4), (6, 3), (7, 4), (8, 2)}

(iii) h: {2, 3, 4, 5} → {7, 9, 11, 13} with

h = {(2, 7), (3, 9), (4, 11), (5, 13)}

If, show that f o f(x) = x, for all. What is the inverse of f?

Find gof and fog, if

(i) 

(ii) 

Let f, g and h be functions from R to R. Show that

Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.