If the function $f: R \rightarrow R$ is defined by $f(x)=|x|(x-\sin x)$, then which of the following statements is TRUE?
- $f$ is one-one, but NOT onto
- $f$ is onto, but NOT one-one
- $f$ is BOTH one-one and onto
- $f$ is NEITHER one-one NOR onto
The Correct Option is C
Solution and Explanation
The Correct Option is (C): $f$ is BOTH one-one and onto
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Concepts Used:
Cartesian Products of Sets
Cartesian products of sets here are explained with the help of an example. Consider A and B to be the 2 sets such that A is a set of 3 colors of tables and B is a set of 3 colors of chairs objects, i.e.,
A = {red, blue, purple}
B = {brown, green, yellow},
Now let us find the number of pairs of colored objects that we can make from a set of tables and chairs in various combinations. They can be grouped as given below:
(red, brown), (red, green), (red, yellow), (blue, brown), (blue, green), (blue, yellow), (purple, brown), (purple, green), (purple, yellow)
There are 9 such pairs in the Cartesian product since 3 elements are there in each of the defined sets A and B. The above-ordered pairs shows the definition for the Cartesian product of sets given. This product is resembled by “A × B”.