Let R be the set of all real numbers and f : R → R be a continuous function. Suppose | f ( x ) - f ( y ) | ≥ | x - y | for all real numbers x and y . Then

f is one-one, but need not be onto

f is onto, but need not be one-one

f need not be either one-one or onto

Show that f ( x ) = 1 , x > 0 0 , x = 0 - 1 , x < 0 is not one-one.

Given that, f ( x ) = 1 , x > 0 0 , x = 0 - 1 , x < 0 f ( x ) = 1 for all x ∈ 0 , ∞ f x = - 1 for all x ∈ - ∞ , 0 So, f x is many-one and not a one-one function.

Let f : R → R be defined by f ( x ) = x 2 - x 2 1 + x 2 for all x ∈ R . Then,

f is one-one but not onto mapping

f is onto but not one-one mapping

The domain of the definition of the function f x = 1 4 - x 2 + l o g 10 x 3 - x is:

- 2 , - 1 ∪ - 1 , 0 ∪ 2 , ∞

Let S , T , U be three non-void sets and f : S → T , g : T → U be so that g o f : S → U is surjective. Then,

g is surjective, f may not be so

f is surjective, g may not be so

f and g both may not be surjective

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Let $f : R \to R$ be defined by $f (x) = x^{4}$ $,$ then

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Important Questions on Relations and Functions

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Mathematics>Algebra>Relations and Functions>Types of Functions

Let

f : [0, 1] \to [- 1, 1]

and

g : [- 1, 1] \to [0, 2]

be two functions such that

g

is injective and

g o f : [0, 1] \to [0, 2]

is surjective. Then,

HARD

Mathematics>Algebra>Relations and Functions>Types of Functions

If the function

f : R \to R

is defined by

f (x) = |x| (x - \sin x)

, then which of the following statements is TRUE?

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Mathematics>Algebra>Relations and Functions>Types of Functions

Let

A

and

B

be finite sets and

P_{A}

and

P_{B}

, respectively denote their power sets. If

P_{B}

has

112

elements more than those in

P_{A}

, then the number of functions from

A

B

which are injective is

HARD

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

R

be the set of all real numbers and

f : R \to R

be a continuous function. Suppose

| f (x) - f (y) | \geq | x - y |

for all real numbers

x

and

y .

Then

EASY

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

f : [2, \infty) \to R

be the function defined by

f (x) = x^{2} - 4 x + 5

, then the range

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Mathematics>Algebra>Relations and Functions>Types of Functions

Let

f : [0, 1] \to R

be an injective continuous function that satisifes the condition

- 1 < f (0) < f (1) < 1

Then, the number of functions

g : [- 1, 1] \to [0, 1]

such that

(g o f) (x) = x

for all

x \in [0, 1]

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

A = [- 1, 1]

and

f : A \to A

be defined as

f (x) = x |x|

for all

x \in A

, then

f (x)

EASY

Mathematics>Algebra>Relations and Functions>Types of Functions

Show that

f (x) = \{\begin{matrix} 1, & x > 0 \\ 0, & x = 0 \\ - 1, & x < 0 \end{matrix}

is not one-one.

HARD

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

f : R \to R

be defined by

f (x) = \frac{|x| - 1}{|x| + 1},

then

f

HARD

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

f : (- \frac{π}{2}, \frac{π}{2}) \to R

be given by

f (x) = {(\log (\sec x + \tan x))}^{3} .

Then

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Mathematics>Algebra>Relations and Functions>Types of Functions

Let

f : R \to R

be defined by

f (x) = x^{2} - \frac{x^{2}}{1 + x^{2}}

for all

x \in R .

Then,

EASY

Mathematics>Algebra>Relations and Functions>Types of Functions

The domain of the definition of the function

f (x) = \frac{1}{4 - x^{2}} + {l o g}_{10} (x^{3} - x)

is:

HARD

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

S, T, U

be three non-void sets and

f : S \to T, g : T \to U

be so that

g o f : S \to U

is surjective. Then,

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Functions

Let $f : R \to R$ be a continuous function such that $f (x^{2}) = f (x^{3})$ for all $x \in R$ . Consider the following statements.

I. $f$ is an odd function.

II. $f$ is an even function.

III. $f$ is differentiable everywhere.

Then,

EASY

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

f : R \to R

defined as

f (x) = x^{3}

, then

f

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Mathematics>Algebra>Relations and Functions>Types of Functions

g

is the inverse of a function

f

and

f^{'} (x) = \frac{1}{1 + x^{5}},

then

g^{'} (x)

is equal to

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Mathematics>Algebra>Relations and Functions>Types of Functions

Function

f : R \to R, f (x) = 5 x + 7

is _____

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Mathematics>Algebra>Relations and Functions>Types of Functions

A = {x | x \in N, x \leq 5}, B = \{x | x \in Z, x^{2} - 5 x + 6 = 0\},

then the number of onto functions from

A

B

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Mathematics>Algebra>Relations and Functions>Types of Functions

Let

f

and

g

be differentiable functions on

R

such that

f o g

is the identity function. If for some

a, b \in R, g^{'} (a) = 5

and

g (a) = b,

then

f^{'} (b)

is equal to:

EASY

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

f : R \to R

be defined as

f (x) = 3 x .

Then

Let f :R→R be defined by fx=x4 , then

Important Questions on Relations and Functions

Learn and Prepare for any exam you want !

Let $f : R \to R$ be defined by $f (x) = x^{4}$ $,$ then