10th Maths Unit 1 Exercise 1.4: Types of Functions Solutions

Exercise 1.4: Types of Functions - Problem Questions with Answer, Solution | Mathematics

Question 1

Determine whether the graph given below represent functions. Give reason for your answers concerning each graph.

Solution:

Question 2

Let \(f : A \rightarrow B\) be a function defined by \(f(x) = \frac{x}{2} - 1\), where \(A = \{2, 4, 6, 10, 12\}\), \(B = \{0, 1, 2, 4, 5, 9\}\). Represent \(f\) by

• (i) set of ordered pairs;
• (ii) a table;
• (iii) an arrow diagram;
• (iv) a graph

Solution:

Question 3

Represent the function \(f = \{(1, 2), (2, 2), (3, 2), (4, 3), (5, 4)\}\) through

• (i) an arrow diagram
• (ii) a table form
• (iii) a graph

Solution:

Question 4

Show that the function \(f: \mathbb{N} \rightarrow \mathbb{N}\) defined by \(f(x) = 2x - 1\) is one-one but not onto.

Solution:

Question 5

Show that the function \(f: \mathbb{N} \rightarrow \mathbb{N}\) defined by \(f(m) = m^2 + m + 3\) is one-one function.

Solution:

Question 6

Let \(A = \{1, 2, 3, 4\}\) and \(B = \mathbb{N}\). Let \(f: A \rightarrow B\) be defined by \(f(x) = x^3\) then,

• (i) find the range of \(f\)
• (ii) identify the type of function

Solution:

Question 7

In each of the following cases state whether the function is bijective or not. Justify your answer.

• (i) \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x) = 2x + 1\)
• (ii) \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x) = 3 - 4x^2\)

Solution:

Question 8

Let \(A = \{-1, 1\}\) and \(B = \{0, 2\}\). If the function \(f: A \rightarrow B\) defined by \(f(x) = ax + b\) is an onto function? Find \(a\) and \(b\).

Solution:

Question 9

If the function \(f\) is defined by $$ f(x) = \begin{cases} x+2 & \text{if } x > 1 \\ 2 & \text{if } -1 \le x \le 1 \\ x-1 & \text{if } -3 < x < -1 \end{cases} $$ find the values of

• (i) \(f(3)\)
• (ii) \(f(0)\)
• (iii) \(f(-1.5)\)
• (iv) \(f(2) + f(-2)\)

Solution:

Question 10

A function \(f: [-5, 9] \rightarrow \mathbb{R}\) is defined as follows: $$ f(x) = \begin{cases} 6x+1 & \text{if } -5 \le x < 2 \\ 5x^2-1 & \text{if } 2 \le x < 6 \\ 3x-4 & \text{if } 6 \le x \le 9 \end{cases} $$ Find

• (i) \(f(-3) + f(2)\)
• (ii) \(f(7) - f(1)\)
• (iii) \(2f(4) + f(8)\)
• (iv) \(\frac{2f(-2) - f(6)}{f(4) + f(-2)}\)

Solution:

Question 11

The distance \(S\) an object travels under the influence of gravity in time \(t\) seconds is given by \(S(t) = \frac{1}{2}gt^2 + at + b\) where, (\(g\) is the acceleration due to gravity), \(a\), \(b\) are constants. Check if the function \(S(t)\) is one-one.

Solution:

Question 12

The function ‘\(t\)’ which maps temperature in Celsius (\(C\)) into temperature in Fahrenheit (\(F\)) is defined by \(t(C) = F\) where \(F = \frac{9}{5}C + 32\). Find,

• (i) \(t(0)\)
• (ii) \(t(28)\)
• (iii) \(t(-10)\)
• (iv) the value of \(C\) when \(t(C) = 212\)
• (v) the temperature when the Celsius value is equal to the Farenheit value.

Solution:

Answers

1. (i) Not a function (ii) function (iii) Not a function (iv) function

2. (i) $\{(2,0),(4,1),(6,2),(10,4),(12,5)\}$

6. (i) $\{1, 8, 27, 64\}$ (ii) one-one and into function

7. (i) Bijective function (ii) Not bijective function

8. 1,1

9. (i) 5 (ii) 2 (iii) -2.5 (iv) 1

10. (i) 2 (ii) 10 (iii) 178 (iv) -9/17

11. Yes

12. (i) 32°F (ii) 82.4°F (iii) 14°F (iv) 100°C (v) -40°