Let f = {(1, 1), (2, 3), (0, -1), (-1, -3)} be a function from Z to Z defined by \(f(x) = ax+ b\), for some integers a, b. Determine a, b.
Solution and Explanation
f = {(1, 1), (2, 3), (0, -1), (-1, -3)}
f(x) = ax + b
(1, 1) ∈ f
⇒ f(1) = 1
⇒ a x 1 + b = 1
⇒ a + b = 1
(0, -1) ∈ f
⇒ f(0) = -1
⇒ a x 0 + b = -1
⇒ b = -1
On substituting b = -1 in a + b = 1, we obtain a + (-1) = 1 ⇒ a = 1 + 1 = 2.
Thus, the respective values of a and b are 2 and -1.
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Concepts Used:
Relations and functions
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.
Representation of Relation and Function
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.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
