Exercise miscellaneous

The relation f is defined by 2 0 3( ) = 3 3 10 x , xf x x, x  ≤ ≤ ≤ ≤ The relation g is defined by 2 , 0 2( ) 3 , 2 10 x xg x x x  ≤ ≤= ≤ ≤ Show that f is a function and g is not a function.

Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)} Are the following true? (i) f is a relation from A to B (ii) f is a function from A to B. Justify your answer in each case.

-1 -11 1 1tan cos4 21 1

x x   + − − π = −   + + −  , 1 1 x− ≤ ≤ [Hint: Put x = cos 2θ] Solve the following equations:

2tan-1 (cos x ) = tan -1 (2 cosec x) 12. -1 -11 1tan tan ,( 0)1 2 x x xx − = >+

Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a function from Z to Z? Justify your answer.

Let A = {9,10,11,12,13} and let f : A→N be defined by f (n) = the highest prime factor of n. Find the range of f.

sin (tan-1 x), | x | < 1 is equal to

(A) 21 / x / x− (B) 2 / 1 x− (C) 2 / 1 x+ (D) 21 / x / x+

sin-1 (1 - x) - 2 sin-1 x = 2 π , then x is equal to (A) 0, 1 / 2 (B) 1, 1 2 (C) 0 (D) 1

If f (x) = x , find (1 1) (1) (1 1 1) f . - f . - .

Find the domain of the function f (x) 2/2 2 1 8 12 x x x - x + += + .

Find the domain and the range of the real function f defined by f (x) = ( 1)x − .

Find the domain and the range of the real function f defined by f (x) = -1x .

Let 2, :

x    = ∈   +    R be a function from R into R. Determine the range of f.

Let f, g : R→R be defined, respectively by f(x) = x + 1, g(x) = 2 x - 3. Find f + g, f - g and f g .

Let f = {(1,1), (2,3), (0,-1), (-1, -3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.

-1 1 sin 1 sincot 21 sin 1 sin

x x   + + − =  + − −  , 0, 4x π ∈  

Let R be a relation from N to N defined by R = {(a, b) : a, b ∈N and a = b }. Are the following true? (i) (a,a) ∈ R, for all a ∈ N (ii) (a,b) ∈ R, implies (b,a) ∈ R (iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R. Justify your answer in each case.