NCERT Solutions
Class 12 Maths
Relations and Functions
Misc. Ex. Q15
Let A = {−1, 0, 1, 2}, B = {−4, −2, 0, 2} and f, g: A → B be functions defined by f(x) = x2− x, x ∈ A and g(x) = 2 |x – 1/2| − 1, x ∈ A. Are f and g equal? Justify your answer.
(Hint: One may note that two function f: A → B and g: A → B such that f(a) = g(a) ∀ a ∈ A, are called equal functions).
It is given that A = {−1, 0, 1, 2}, B = {−4, −2, 0, 2}.
Also, it is given that f, g: A → B are defined by
f(x) = x2− x, x ∈ A and g(x) = 2 |x – 1/2| − 1, x ∈ A
It is observed that
f(−1) = (−1)2 − (−1) = 1 + 1 = 2
and g(−1) = 2 |(−1) – 1/2| − 1 = 2(3/2) − 1 = 3 − 1 = 2
⇒ f(−1) = g(−1)
f(0) = (0)2 − (0) = 0
and g(0) = 2 |0 – 1/2| − 1 = 2(1/2) − 1 = 1 − 1 = 0
⇒ f(0) = g(0)
f(1) = (1)2 − (1) = 1 − 1 = 0
and g(1) = 2 |(1) – 1/2| − 1 = 2(1/2) − 1 = 1 − 1 = 0
⇒ f(1) = g(1)
f(2) = (2)2 − (2) = 4 − 2 = 2
and g(2) = 2 |(2) – 1/2| − 1 = 2(3/2) − 1 = 3 − 1 = 2
⇒ f(2) = g(2)
So, f(a) = g(a) for all a ∈ A
Hence, the functions f and g are equal.