QBManager

Sets, Relation and Function

117268 If $f (\frac{x + 1}{2 x - 1}) = 2 x, x \in N$ , then the value of $f (2)$ is equal to

1 1

2 4

3 3

4 2

5 5

Explanation:

D Given,
$f (\frac{x + 1}{2 x - 1}) = 2 x$
Putting $x = 1$
$f (\frac{1 + 1}{2 (1) - 1}) = 2 \times 1$
$f (\frac{2}{2 - 1}) = 2$ Hence, $f (2) = 2$

Kerala CEE-2016

Sets, Relation and Function

117227 If $f : A \to B$ and $g : B \to C$ are two functions such that gof : $A \to C$ is a bijections, then which one of the following is always true?

1 f and g are bijections

2

f

is an injection and

g

is a surjection

3

f

is a surjection and

g

is an injection

4

f

is a bijection but

g

is not a bijection

Explanation:

B $gof (y) = z$
$g (f (y)) = z$
$f$ lies range but $f : A \to B$
for $a, y \in A, f (y) \in B$ , let $f (y) = x$
For every element $z \in c$ , we have at $x \in B$ .
Such that
$g (x) = z$
$∴ g$ is onto.
Hence, $f$ is an injection and $g$ is a surjection.

Shift-I

Sets, Relation and Function

117229 Which of the following is true?

1 The composition of function is commutative

2 Every function is invertible

3 If a function

f

is bijective then its inverse

f^{- 1}

need not be bijective

4 The composition of functions is associative

Explanation:

D Consider $f : A \to B$ and $g : B \to C$ be two functions. Then the composition of $f$ and $g$ denoted by gof. is defined as the function gof: $A \to C$ given by go gof $(x) = g f (x)$
Properties of function compositions.
Associative property : Associative property of function composition,
If there are three functions $f, g$ and $h$ , then they are said to be associative if and only is -
$fo (g o h) = (f o g) oh.$
The function composition of two onto function is always onto.
The function composition of one - to - one function is always onto.
The function composition of one- to-one function is always one to one
So, the composition of function is associative is true statement.

JCECE-2018

Sets, Relation and Function

117235 If $f (x) = \frac{4 x + 7}{7 x - 4}$ , then the value of $f {f [f (2)]} =$

1

\frac{35}{39}

2

\frac{2}{3}

3

\frac{3}{2}

4

\frac{39}{35}

Explanation:

C Given,
$f (x) = \frac{4 x + 7}{7 x - 4}$
$f (2) = \frac{8 + 7}{14 - 4} = \frac{15}{10} = \frac{3}{2}$
$f (f (2)) = f (\frac{3}{2}) = \frac{(4 \times \frac{3}{2}) + 7}{(7 \times \frac{3}{2}) - 4} = \frac{6 + 7}{(\frac{21 - 8}{2})} = \frac{13 \times 2}{13} = 2$ So, $f {f [f (2)]} = f (2) = \frac{3}{2}$

MHT-CET 2020

Sets, Relation and Function

117268 If $f (\frac{x + 1}{2 x - 1}) = 2 x, x \in N$ , then the value of $f (2)$ is equal to

1 1

2 4

3 3

4 2

5 5

Explanation:

D Given,
$f (\frac{x + 1}{2 x - 1}) = 2 x$
Putting $x = 1$
$f (\frac{1 + 1}{2 (1) - 1}) = 2 \times 1$
$f (\frac{2}{2 - 1}) = 2$ Hence, $f (2) = 2$

Kerala CEE-2016

Sets, Relation and Function

117227 If $f : A \to B$ and $g : B \to C$ are two functions such that gof : $A \to C$ is a bijections, then which one of the following is always true?

1 f and g are bijections

2

f

is an injection and

g

is a surjection

3

f

is a surjection and

g

is an injection

4

f

is a bijection but

g

is not a bijection

Explanation:

B $gof (y) = z$
$g (f (y)) = z$
$f$ lies range but $f : A \to B$
for $a, y \in A, f (y) \in B$ , let $f (y) = x$
For every element $z \in c$ , we have at $x \in B$ .
Such that
$g (x) = z$
$∴ g$ is onto.
Hence, $f$ is an injection and $g$ is a surjection.

Shift-I

Sets, Relation and Function

117229 Which of the following is true?

1 The composition of function is commutative

2 Every function is invertible

3 If a function

f

is bijective then its inverse

f^{- 1}

need not be bijective

4 The composition of functions is associative

Explanation:

D Consider $f : A \to B$ and $g : B \to C$ be two functions. Then the composition of $f$ and $g$ denoted by gof. is defined as the function gof: $A \to C$ given by go gof $(x) = g f (x)$
Properties of function compositions.
Associative property : Associative property of function composition,
If there are three functions $f, g$ and $h$ , then they are said to be associative if and only is -
$fo (g o h) = (f o g) oh.$
The function composition of two onto function is always onto.
The function composition of one - to - one function is always onto.
The function composition of one- to-one function is always one to one
So, the composition of function is associative is true statement.

JCECE-2018

Sets, Relation and Function

117235 If $f (x) = \frac{4 x + 7}{7 x - 4}$ , then the value of $f {f [f (2)]} =$

1

\frac{35}{39}

2

\frac{2}{3}

3

\frac{3}{2}

4

\frac{39}{35}

Explanation:

C Given,
$f (x) = \frac{4 x + 7}{7 x - 4}$
$f (2) = \frac{8 + 7}{14 - 4} = \frac{15}{10} = \frac{3}{2}$
$f (f (2)) = f (\frac{3}{2}) = \frac{(4 \times \frac{3}{2}) + 7}{(7 \times \frac{3}{2}) - 4} = \frac{6 + 7}{(\frac{21 - 8}{2})} = \frac{13 \times 2}{13} = 2$ So, $f {f [f (2)]} = f (2) = \frac{3}{2}$

MHT-CET 2020

Sets, Relation and Function

117268 If $f (\frac{x + 1}{2 x - 1}) = 2 x, x \in N$ , then the value of $f (2)$ is equal to

1 1

2 4

3 3

4 2

5 5

Explanation:

D Given,
$f (\frac{x + 1}{2 x - 1}) = 2 x$
Putting $x = 1$
$f (\frac{1 + 1}{2 (1) - 1}) = 2 \times 1$
$f (\frac{2}{2 - 1}) = 2$ Hence, $f (2) = 2$

Kerala CEE-2016

Sets, Relation and Function

117227 If $f : A \to B$ and $g : B \to C$ are two functions such that gof : $A \to C$ is a bijections, then which one of the following is always true?

1 f and g are bijections

2

f

is an injection and

g

is a surjection

3

f

is a surjection and

g

is an injection

4

f

is a bijection but

g

is not a bijection

Explanation:

B $gof (y) = z$
$g (f (y)) = z$
$f$ lies range but $f : A \to B$
for $a, y \in A, f (y) \in B$ , let $f (y) = x$
For every element $z \in c$ , we have at $x \in B$ .
Such that
$g (x) = z$
$∴ g$ is onto.
Hence, $f$ is an injection and $g$ is a surjection.

Shift-I

Sets, Relation and Function

117229 Which of the following is true?

1 The composition of function is commutative

2 Every function is invertible

3 If a function

f

is bijective then its inverse

f^{- 1}

need not be bijective

4 The composition of functions is associative

Explanation:

D Consider $f : A \to B$ and $g : B \to C$ be two functions. Then the composition of $f$ and $g$ denoted by gof. is defined as the function gof: $A \to C$ given by go gof $(x) = g f (x)$
Properties of function compositions.
Associative property : Associative property of function composition,
If there are three functions $f, g$ and $h$ , then they are said to be associative if and only is -
$fo (g o h) = (f o g) oh.$
The function composition of two onto function is always onto.
The function composition of one - to - one function is always onto.
The function composition of one- to-one function is always one to one
So, the composition of function is associative is true statement.

JCECE-2018

Sets, Relation and Function

117235 If $f (x) = \frac{4 x + 7}{7 x - 4}$ , then the value of $f {f [f (2)]} =$

1

\frac{35}{39}

2

\frac{2}{3}

3

\frac{3}{2}

4

\frac{39}{35}

Explanation:

C Given,
$f (x) = \frac{4 x + 7}{7 x - 4}$
$f (2) = \frac{8 + 7}{14 - 4} = \frac{15}{10} = \frac{3}{2}$
$f (f (2)) = f (\frac{3}{2}) = \frac{(4 \times \frac{3}{2}) + 7}{(7 \times \frac{3}{2}) - 4} = \frac{6 + 7}{(\frac{21 - 8}{2})} = \frac{13 \times 2}{13} = 2$ So, $f {f [f (2)]} = f (2) = \frac{3}{2}$

MHT-CET 2020

Sets, Relation and Function

117268 If $f (\frac{x + 1}{2 x - 1}) = 2 x, x \in N$ , then the value of $f (2)$ is equal to

1 1

2 4

3 3

4 2

5 5

Explanation:

D Given,
$f (\frac{x + 1}{2 x - 1}) = 2 x$
Putting $x = 1$
$f (\frac{1 + 1}{2 (1) - 1}) = 2 \times 1$
$f (\frac{2}{2 - 1}) = 2$ Hence, $f (2) = 2$

Kerala CEE-2016

Sets, Relation and Function

117227 If $f : A \to B$ and $g : B \to C$ are two functions such that gof : $A \to C$ is a bijections, then which one of the following is always true?

1 f and g are bijections

2

f

is an injection and

g

is a surjection

3

f

is a surjection and

g

is an injection

4

f

is a bijection but

g

is not a bijection

Explanation:

B $gof (y) = z$
$g (f (y)) = z$
$f$ lies range but $f : A \to B$
for $a, y \in A, f (y) \in B$ , let $f (y) = x$
For every element $z \in c$ , we have at $x \in B$ .
Such that
$g (x) = z$
$∴ g$ is onto.
Hence, $f$ is an injection and $g$ is a surjection.

Shift-I

Sets, Relation and Function

117229 Which of the following is true?

1 The composition of function is commutative

2 Every function is invertible

3 If a function

f

is bijective then its inverse

f^{- 1}

need not be bijective

4 The composition of functions is associative

Explanation:

D Consider $f : A \to B$ and $g : B \to C$ be two functions. Then the composition of $f$ and $g$ denoted by gof. is defined as the function gof: $A \to C$ given by go gof $(x) = g f (x)$
Properties of function compositions.
Associative property : Associative property of function composition,
If there are three functions $f, g$ and $h$ , then they are said to be associative if and only is -
$fo (g o h) = (f o g) oh.$
The function composition of two onto function is always onto.
The function composition of one - to - one function is always onto.
The function composition of one- to-one function is always one to one
So, the composition of function is associative is true statement.

JCECE-2018

Sets, Relation and Function

117235 If $f (x) = \frac{4 x + 7}{7 x - 4}$ , then the value of $f {f [f (2)]} =$

1

\frac{35}{39}

2

\frac{2}{3}

3

\frac{3}{2}

4

\frac{39}{35}

Explanation:

C Given,
$f (x) = \frac{4 x + 7}{7 x - 4}$
$f (2) = \frac{8 + 7}{14 - 4} = \frac{15}{10} = \frac{3}{2}$
$f (f (2)) = f (\frac{3}{2}) = \frac{(4 \times \frac{3}{2}) + 7}{(7 \times \frac{3}{2}) - 4} = \frac{6 + 7}{(\frac{21 - 8}{2})} = \frac{13 \times 2}{13} = 2$ So, $f {f [f (2)]} = f (2) = \frac{3}{2}$

MHT-CET 2020