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Question
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 in A`. Are f and g equal?
Justify your answer. (Hint: One may note that two functions f: A → B and g: A → B such that f(a) = g(a) ∀ a ∈ A are called equal functions).
Solution
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 in A`.
It is observed that:
`f(-1) = (1^2) - (-1) = 1+1 = 2`
`g(-1) = 2|(-1)-1/2| - 1`
`= 2(3/2) - 1`
= 3 -1
=2
=> f(-1) = g(-1)
f(0) = (0)^2 - 0 = 0
`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
`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
`g(2) = 2|2-1/2| - 1`
` = 2(3/2)-1 `
= 3 -1
= 2
`=> f(2) = g(2)`
:. f(a) = g(a) ∀ a ∈ A
Hence, the functions f and g are equal.
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