Solutions of chapter 2. Functions | RS Aggarwal - Mathematics

1

Define a function. What do you mean by the domain and range of a function? Give examples.

2

Define each of the following:

(i) injective function

(ii) surjective function

(iii) bijective function

(iv) many - one function

(v) into function

Give an example of each type of functions.

3

Give an example of a function which is

(i) one - one but not onto

(ii) one - one and onto

(iii) neither one - one nor onto

(iv) onto but not one - one.

4

Let f : R → R be defined by

Find (i) f(2) (ii) f(4) (iii) f( - 1) (iv) f( - 3).

5

Show that the function f: R → R : f(x) = 1 + x² is many - one into.

6

Show that the function f : R → R : f(x) = x⁴ is many - one and into.

7

Show that the function f: R → R : f(x) = x⁵ is one - one and onto.

8

Let and Show that each one of f and g is one - one but (f + g) is not one - one.

9

Show that the function

(i) f : N → N : f(x) = x² is one - one into.

(ii) f : Z → Z : f(x) = x² is many - one into

10

Show that the function

(i) f : N → N : f(x) = x³ is one - one into

(ii) f : Z → Z : f(x) = x³ is one - one into

11

Show that the function f : R → R : f(x) = sin x is neither one - one nor onto.

12

Prove that the function f : N → N : f(n) = (n² + n + 1) is one - one but not onto.

13

Show that the function f: N → Z, defined by

is both one - one and onto.

14

Find the domain and range of the function

F : R → R : f(x) = x² + 1.

15

Which of the following relations are functions? Give reasons. In case of a function, find its domain and range.

(i) f = {( - 1, 2), (1, 8), (2, 11), (3, 14)}

(ii) g = {(1, 1), (1, - 1), (4, 2), (9, 3),
(16, 4)}

(iii) h = {(a, b), (b, c), (c, b), (d, c)}

16

Find the domain and range of the real function, defined by Show that f is many - one.

17

Show that the function

is many - one into.

Find (i) (ii) (iii)

(iv)

1

Let A = {1, 2, 3, 4}. Let f : A → A and g : A → A,

defined by f = {(1, 4), (2, 1), (3,3),(4, 2)} and g = {(1, 3), (2, 1), (3, 2), (4, 4)}.

Find (i) g of (ii) f o g (iii) f o f.

2

Let f : {3, 9, 12} → {1, 3, 4} and g : {1, 3, 4, 5} → {3, 9} be

defined as f = {(3, 1), (9, 3), (12, 4)} and

g = {(1, 3), (3, 3), (4, 9), (5, 9)}.

Find (i) (g o f) (ii) (f o g).

3

Let f : R → R : f(x) = x² and g : R → R : g(x) = (x + 1).

Show that (g o f) ≠ (f o g).

4

Let f : R → R : f(x) = (2x + 1) and g : R → R : g(x) = (x² - 2).

Write down the formulae for

(i) (g o f) (ii) (f o g)

(iii) (f o f) (iv) (g o g)

5

Let f : R → R : f(x) = (x² + 3x + 1) and g: R → R : g(x) = (2x - 3). Write down the formulae for

(i) g o f

(ii) f o g

(iii) g o g

6

Let f : R → R : f(x) = |x|, prove that f o f = f.

7

Let f : R → R : f(x) = x^2, g : R → R : g(x) = tan x

and h : R → R : h(x) = log x.

Find a formula for h o (g o f).

Show that [h o (g o f)]

8

Let f : R → R : f(x) (2x - 3) and

Show that (f o g) = I_R = (g o f).

9

Let f : Z → Z : f(x) = 2x. Find g : Z → Z : g o f = I_Z.

10

Let f : N → N : f(x) = 2x, g : N → N : g(y) = 3y + 4 and h : N → N : h(z) = sin z. Show that h o (g o f ) = (h o g) o f.

11

If f be a greatest integer function and g be an absolute value function, find the value of

12

Let f : R → R : f(x) = x² + 2 and find f o g and g o f and hence find (f o g) (2) and (g o f) (-3).

1

Prove that the function f: R → R : f(x)= 2x is one-one and onto.

2

Prove that the function f: N → N : f (x)=3x is one-one and into.

3

Show that the function f : R → R : f (x) = x² is neither one-one nor onto.

4

Show that the function f : N → N : f (x) = x² is one-one and into.

5

Show that the function f : R → R : f (x) = x⁴ is neither one-one nor onto.

6

Show that the function f : Z → Z : f (x) = x³ is one-one and into.

7

Let R₀ be the set of all nonzero real numbers. Then, show that the function is one-one and onto.

8

Show that the function f : R → R : f(x) = 1 + x² is many-one into.

9

Let be an invertible function. Find f^-1.

10

Let f : R → R : f(x) = 10x + 3. Find f^-1.

11

Show that f is many-one and into.

12

Let f (x) = x + 7 and g(x) = x -7, x ∈ R. Find (f o g) (7).

13

Let f : R → R and g : R → R defined by f(x) = x² and g(x) = (x + 1). Show that g o f ≠ f o g.

14

Let f : R → R : f(x) = (3 - x³)^1/3. Find f o f.

15

Let f : R → R : f(x) = 3x + 2, find f{f(x)}.

16

Let f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)). Write down g o f.

17

Let A = {1, 2, 3, 4} and f = {(1, 4), (2, 1) (3, 3), (4, 2)}. Write down (f o f).

18

Let f(x) = 8x³ and g(x) = x^1/3. Find g o f and f o g.

19

Let f : R → R : f(x) = 10x + 7. Find the function g : R → R : g o f = f o g = I_g.

20

Let A = {1, 2, 3), B = {4, 5, 6, 7) and let f = {(1, 4), (2,5), (3, 6)} be a function from A to B. State whether f is one-one.

1

Let A = {2, 3, 4, 5} and B = {7, 9, 11, 13}, and

let f = {(2, 7), (3, 9), (4, 11), (5, 13)}.

Show that f is invertible and find f^-1.

2

Show that the function f : R → R : f(x) = 2x + 3 is invertible and find f^-1.

3

Let f : Q → Q : f(x) = 3x —4. Show that f is invertible and find f^-1.

4

Let . Show that f is invertible and find f^-1.

5

If show that (f o f) (x) = x for all

Hence, find f^-1.

6

Show that the function f on defined as is one-one and onto. Hence, find f^-1.

7

Show that the function f on into itself, defined by is one-one and onto. Hence, find f^-1.

8

Let R₊ be the set of all positive real numbers. show that the function f : R₊→ [-5, ∞]: f(x) = (9x² + 6x – 5) is invertible. Find f^-1.

9

Let f : N → R : f(x) = 4x² + 12x + 15. Show that f: N → range (f) is invertible. Find f^-1.

10

Let A = R – {2} and B = R – {1}. If show that f is one-one and onto. Hence, find f^-1.

11

Let f and g be two functions from R into R, defined by f(x) = |x| + x and g(x) = |x| - x for all x ∈ R. Find f o g and g o f.

1