Let f, g:R\(\to\)R be defined, respectively by f(x) = x+1, g(x) = 2x-3. Find f+g, f-g and \(\frac fg\).
Solution and Explanation
f, g: R\(\to\)R is defined as f(x) = x+1, g(x) = 2x-3
(f+g)(x) = f(x) + g(x) = (x+1) + (2x-3) = 3x-2
∴ (f+g)(x) = 3x-2
(f-g)(x) = f(x) - g(x) = (x+1) - (2x-3) = x+1-2x+3 = - x+4
∴ (f-g)(x) = -x+4
(\(\frac fg\))(x) = \(\frac {f(x)}{g(x)}\), g(x)≠0,x∈R
(\(\frac fg\))(x) = \(\frac {x+1}{2x-3}\), 2x-3≠0 or 2x≠3
(\(\frac fg\))(x) = \(\frac {x+1}{2x-3}\), x≠\(\frac 32\)
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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
