# Solution of Chapter 3. Functions (RS Aggarwal - Mathematics Book)

## Exercise 3A

1

Define a function as a set of ordered pairs.

2

Define a function as a correspondence between two sets.

3

What is the fundamental difference between a relation and function? Is every relation a function?

4

Let X = {1, 2, 3, 4,}, Y = {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 X to Y (ii) F is a function from X to Y. Justify your answer in following true?

5

Let X = {-1, 0, 3, 7, 9} and f : X R : f(x) x3 + 1. Express the function f as set of ordered pairs.

6

Let A = {–1, 0, 1, 2} and B = {2, 3, 4, 5}. Find which of the following are function from A to B. Give reason.

(i) f = {(–1, 2), (-1, 3), (0, 4), 1,5)}

(ii) g = {(0, 2), (1, 3), (2, 4)}

(iii) h = {(-1, 2), (0, 3), (1, 4), (2, 5)}

7

Let A = {1, 2} and B = {2, 4, 6}. Let f = {(x, y) : x ϵ A, y ϵ B and y > 2x + 1}. Write f as a set of ordered pairs. Show that f is a relation but not a function from A to B.

8

Let A = {0, 1, 2} and B = {3, 5, 7, 9}. Let f = {(x, y) : x ϵ A, y ϵ B and y = 2x + 3}. Write f as a set of ordered pairs. Show that f is function from A to B. Find dom (f) and range (f).

9

Let A = {2, 3, 5, 7} and B = {3, 5, 9, 13, 15}. Let f = {(x, y) : x ϵ A, y ϵ B and y = 2x – 1}. Write f in roster form. Show that f is a function from A to B. Find the domain and range of f.

10

Let g = {(1, 2), (2, 5), (3, 8), (4, 10), (5, 12), (6, 12)}. Is g a function? If yes, its domain range. If no, give reason.

11

Let f = {(0, -5), (1, -2), (3, 4), (4, 7)} be a linear function from Z into Z. Write an expression for f.

12
13

If f(x) = x2, find the value of .

14

Let X = {12, 13, 14, 15, 16, 17} and f : A Z : f(x) = highest prime factor of x. Find range (f)

15

Let R+ be the set of all positive real numbers. Let f : R+ R : f(x) = logex. Find

(i) range (f)

(ii) {x : x ϵ R+ and f(x) = -2}.

(iii) Find out whether f(x + y) = f(x). f(y) for all x, y ϵ R.

16

Let f : R R : f(x) =2x. Find

(i) range (f)

(ii) {x : f(x) = 1}.

(iii) Find out whether f(x + y) = f(x). f(y) for all x, y ϵ R.

17

Let f : R R : f(x) =x2 and g : C C: g(x) =x2, where C is the set of all complex numbers. Show that f ≠ g.

18

f, g and h are three functions defined from R to R as following:

(i) f(x) = x2

(ii) g(x) = x2 + 1

(iii) h(x) = sin x

That, find the range of each function.

19

Let f : R R : f(x) = x2 + 1. Find

(i) f–1 {10}

(ii) f–1 {–3}.

20

The function is the formula to convert x °C to Fahrenheit units. Find

(i) F(0),

(ii) F(–10),

(iii) The value of x when f(x) = 212.

Interpret the result in each case.

## Exercise 3B

1

If f(x) = x2 – 3x + 4 and f(x) = f(2x + 1), find the values of x.

2

(i) (ii) 3

If then show that 4

If then show that f{f(x)} = x.

5

If f(x) = and then prove that f{(x)} = , when it is given that .

6

If then show that f [f{f(x)}] = x.

7

If then show that f(tanθ) = sin 2θ.

8

If , prove that x = f(y).

## Exercise 3C

1

Find the domain of each of the following real function.

(i) (ii) (iii) (iv) 2

Find the domain and the range of each of the following real function: 3

Find the domain and the range of each of the following real function: 4

Find the domain and the range of each of the following real function: 5

Find the domain and the range of each of the following real function: 6

Find the domain and the range of each of the following real function: 7

Find the domain and the range of each of the following real function: f(x) = 8

Find the domain and the range of each of the following real function: f(x) = 9

Find the domain and the range of each of the following real function: 10

Find the domain and the range of each of the following real function: 11

Find the domain and the range of each of the following real function: 12

Find the domain and the range of each of the following real function: f(x) = 1 – |x – 2|

13

Find the domain and the range of each of the following real function: 14

Find the domain and the range of each of the following real function: 15

Find the domain and the range of each of the following real function: ## Exercise 3D

1

Consider the real function f: R R: f(x) = x + 5 for all

x ϵ R. Find its domain and range. Draw the graph of this function.

2

Consider the function f: R R, defined by Write its domain and range. Also, draw the graph of f(x).

3

Find the domain and the range of the square root function,

f: R+ U {0} R f(x) = for all non-negative real numbers.

Also, draw its graph.

4

Find the domain and the range of the cube root function,

f: R R: f(x) = x1/3 for all x ϵ R. Also, draw its graph.

## Exercise 3E

1

Let f : R R : f(x) = x + 1 and g : R R : g(x) = 2x – 3.Find

(i) (f + g) (x)

(ii) (f – g) (x)

(iii) (fg) (x)

(iv)(f/g) (x)

2

Let f : R R : f(x) = 2x + 5 and g : R R : g(x) = x2 + x.

Find

(i) (f + g) (x)

(ii) (f – g) (x)

(iii) (fg) (x)

(iv) (f/g)(x)

3

. Let f: R R: f(x) = x3 + 1 and g: R R: g(x) = (x + 1). Find:

(i) (f + g) (x)

(ii) (f – g) (x)

(iii) (1/f) (x)

(iv) (f/g) (x)

4

. Let f: R R; f(x)= ,where c is a constant. Find

(i) (cf) (x)

(ii) (c2 f) (x)

(iii) 5

. Let f:(2, ∞) R: f(x) = and g: (2, ∞) R: g(x) = Find:

(i) (f + g) (x)

(ii) (f - g) (x)

(iii) (fg) (x)

## Exercise 3F

1

Find the set of values for which the function f(x) = 1 – 3x and g(x) = 2x2 – 1 are equal.

2

Find the set of values for which the function f(x) = x + 3 and g(x) = 3x2 – 1 are equal.

3

Let X = {–1, 0, 2, 5} and f : X R Z: f(x) = x3 + 1. Then, write f as a set of ordered pairs.

4

Let A = {–2, –1, 0, 2} and f : A Z: f(x) = x2 – 2x – 3. Find f(A).

5

Let f : R R : f(x) = x2.

Determine (i) range (f) (ii) {x : f(x) = 4}

6

Let f : R R : f(x) = x2 + 1. Find f–1 {10}.

7

Let f : R+ R : f(x) = loge x. Find {x : f(x) = –2}.

8

Let A = {6, 10, 11, 15, 12} and let f : A N : f(n) is the highest prime factor of n. Find range (f).

9

Find the range of the function f(x) = sin x.

10

Find the range of the function f(x) = |x|.

12

If then find dom (f) and range (f).

13

Let f = {(1, 6), (2, 5), (4, 3), (5, 2), (8, –1), (10, –3)} and g = {(2, 0), (3, 2), (5, 6), (7, 10), (8, 12), (10, 16)}.

Find (i) dom (f + g) (ii) dom .

14

If , find the value of .

15

If , where x ≠ –1 and f{f(x)} = x for x ≠ –1 then find the value of k.

16

Find the range of the function, .

17

Find the domain of the function, f(x) = log |x|.

18

If = for all x ϵ R – {0} then write an expression for f(x).

19

Write the domain and the range of the function, .

Write the domain and the range of the function, f(x) = .