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. 

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

(f + g)oh = foh + goh

(f . g)oh = (foh) . (goh) 

 Find gof and fog, if

(i) f(x) = | x | and g(x) = | 5x – 2 |
(ii) f(x) = 8x3 and g(x) = x^1/3 .

 If f(x) = , show that fof(x) = x, for all x ≠ 2/3. What is the inverse of f ?

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)}

Show that f : [–1, 1] → R, given by is 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.,)

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

 Consider f : R₊ → [4, ∞) given by f(x) = x² + 4. Show that f is invertible with the inverse f^–1 of f given by , where R₊ is the set of all non-negative real numbers.

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

 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) = 1Y(y) = fog2(y). Use one-one ness of f). 

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

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

 If f: R → R be given by f(x) = , then fof(x) is
(A)  

(B)  x³

(C)  x

(D)  (3 – x³).

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

(A)  

(B) 

(C)  

(D)