Find the range of each of the following functions.
(i) f(x) = 2 - 3x, x ∈ R, x> 0.
(ii) f(x) = x2+ 2, x, is a real number.
(iii) f(x) = x, x is a real number
(i) f(x) = 2 -3x, x ∈ R, x> 0
The values of f(x) for various values of real numbers x> 0 can be written in the tabular form as
x | 0.01 | 0.1 | 0.9 | 1 | 2 | 2.5 | 4 | 5 | ... |
f(x) | 1.97 | 1.7 | -0.7 | -1 | -4 | -5.5 | -10 | -13 | ... |
Thus, it can be clearly observed that the range of fis the set of all real numbers less than 2.
i.e., range of f= (-, 2)
Alter:
Let x > 0
⇒3x > 0
⇒ 2-3x< 2
⇒ f(x) < 2
∴Range of f = (-, 2)
(ii) f(x) = x2+ 2, x, is a real number
The values of f(x) for various values of real numbers xcan be written in the tabular form as
x | 0 | ±0.3 | ±0.8 | ±1 | ±2 | ±3 | ..… | |
f(x) | 2 | 2.09 | 2.64 | 3 | 6 | 11 | ..… |
Thus, it can be clearly observed that the range of fis the set of all real numbers greater than 2.
i.e., range of f= [2, ∞)
Alter:
Let x be any real number.
Accordingly,
x2 ≥0
⇒ x2+ 2 ≥0 + 2
⇒ x2+ 2 ≥2
⇒ f(x) ≥2
∴ Range of f = [2, )
(iii) f(x) = x, x is a real number
It is clear that the range of fis the set of all real numbers.
∴ Range of f = R
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is:
Give reasons for the following.
(i) King Tut’s body has been subjected to repeated scrutiny.
(ii) Howard Carter’s investigation was resented.
(iii) Carter had to chisel away the solidified resins to raise the king’s remains.
(iv) Tut’s body was buried along with gilded treasures.
(v) The boy king changed his name from Tutankhaten to Tutankhamun.
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.