Let \(f={(x,\frac {x^2}{1+x^2}):x∈R}\) be a function from R into R. Determine the range of f.
Solution and Explanation
\(f={(x,\frac {x^2}{1+x^2}):x∈R}\)
= \({(0,0),(±0.5,\frac 15),(±1,\frac 12),(±1.5,\frac {9}{13}),(±2,\frac {4}{5}),(3,\frac {9}{10}),(4,\frac{16}{17}), ......}\)
The range of f is the set of all second elements. It can be observed that all these elements are greater than or equal to 0 but less than 1.
[Denominator is greater numerator]
Thus, range of f= [0, 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
