Page 19 - Relations and Functions 19.10.06.pmd

RELATIONS AND FUNCTIONS 19

2
9. Consider f : R → [– 5, ∞) given by f (x) = 9x + 6x – 5. Show that f is invertible
+
⎛ ( y + ) 6 − 1 ⎞
with f (y) = ⎜ –1 ⎟ .
⎝ 3 ⎠
10. Let f : X → Y be an invertible function. Show that f has unique inverse.
(Hint: suppose g and g are two inverses of f. Then for all y ∈ Y,
1 2
fog (y) = 1 (y) = fog (y). Use one-one ness of f).
1 Y 2
11. Consider f : {1, 2, 3} → {a, b, c} given by f (1) = a, f (2) = b and f (3) = c. Find
f and show that (f ) = f.
–1 –1
–1
–1
12. Let f: X → Y be an invertible function. Show that the inverse of f is f, i.e.,
(f ) = f.
–1 –1
1
13. If f: R → R be given by f (x) = (3 x− 3 3
) , then fof (x) is
1
(A) x 3 (B) x 3 (C) x (D) (3 – x ).
3
⎧ 4⎫ 4x
⎨
14. Let f : R – − ⎬ → R be a function defined as f (x) = . The inverse of
⎩ 3 ⎭ 3x + 4
⎧ 4⎫
⎨
f is the map g : Range f → R – − ⎬ given by
⎩ 3⎭
3y 4y
() =
() =
(A) gy (B) gy
34y− 43y−
4y 3y
() =
() =
(C) gy (D) gy
34y− 43y−
1.5 Binary Operations
Right from the school days, you must have come across four fundamental operations
namely addition, subtraction, multiplication and division. The main feature of these
operations is that given any two numbers a and b, we associate another number a + b

a
or a – b or ab or , b ≠ 0. It is to be noted that only two numbers can be added or
b
multiplied at a time. When we need to add three numbers, we first add two numbers
and the result is then added to the third number. Thus, addition, multiplication, subtraction