Let f(x) be defined for all x > 0 and be continuous. If f(x) satisfies  for all x, y and f(e)=1 then

a) f(x) is bounded

b)

c) x f(x) → 1 as x → 0

d) f(x) = lnx

If , for every real number x, then the minimum value of f

a) does not exist because f is unbounded

b) is not attained even though f is bounded

c) is equal to 1

d) is equal to –1

If  then

a)

b)

c)

d) f and g cannot be determined

The number of values of x where the function  attains its maximum is

a) 0

b) 1

c) 2

d) infinite

The function  is defined by then  is

a)

b)

c)

d) None of these

The domain of the definition of the function y given by the equation  is

a) 0 < x < 1

b) 0 ≤ x ≤ 1

c) ∞ < x ≤ 0

d) ∞ < x ≤ 1

The domain of definition of  is

a)  

b)  

c)  

d)  

If f : [1, ∞) → [2, ∞) is given by  then  equals

a)

b)

c)

d)

Let f(x) =   , x ≠  then for what value of α, f(f(x)) = x

a)

b)

c)

d)

Suppose  for x ≥ . If g(x) is the function whose graph is the reflection of f(x) with respect to the line y = x then g(x) equals

a)

b)

c)

d)

Let f : ℝ → ℝ be defined by f(x) = 2x + sinx for all x  ℝ. Then f is

a) One to one and onto

b) One to one but not onto

c) Onto but not one to one

d) Neither one to one nor onto

If  and  then f is

a) One-one and onto

b) One-one but not onto

c) Onto but not one-one

d) Neither one-one nor onto

Domain of definition of the function   for real values of x is

a)

b)

c)

d)

Range of    ;   x  ℝ is

a) (1, ∞)

b)

c)

d)

If
and
Then f – g is

a) Neither one to one nor onto

b) One to one and onto

c) One to one and into

d) Many one and onto

Subjective problems
Let .  Find all real values of x for which y takes real values.

a) [− 1, 2)

b)  [3, ∞)

c) [− 1, 2) ∪ [3, ∞)

d) None of the above

Let f be a one–one function with domain {x, y, z} and range {1, 2, 3}. It is given that exactly one of the following statements is true and remaining statements are false f (1) = 1, f (y) ≠ 1, f (z) ≠ 2. Determine  

Let R be the set of real numbers and f : R → R be such that for all x and y in R, . Prove that f(x) is constant.

Find the natural number a for which  where the function f satisfies the relation f(x + y) = f(x) f(y) for all natural numbers x and y and further f(1) = 2.

Let {x} and [x] denote the fractional and integral part of a real number x respectively. Solve 4{x} = x + [x]

Let  where A, B, C are real numbers. Prove that if f(n) is an integer whenever n is an integer, then the numbers 2A, A + B and C are all integers. Conversely prove that if the numbers 2A, A + B and C all integers then f(n) is an integer whenever n is an integer.

If where a > 0 and n is a positive integer then f(f(x)) = x.

a) True

b) False

If f₁(x) and f₂(x) are defined by domains D₁ and D₂ respectively then f₁(x) + f₂(x) is defined as on D₁ ⋂ D₂

a) True

b) False