Question

Mapping $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ which is defined as $\mathrm{f}(\mathrm{x})=\cos \mathrm{x}, \mathrm{x} \in \mathrm{R}$ will be
A.Neither one-one nor onto
B.One-One
C.Onto
D.One-One onto

Answer 1

Hint: In mathematics, a map is often used as a synonym for a function, but may also refer to some generalizations. Originally, this was an abbreviation of mapping, which often refers to the action of applying a function to the elements of its domain. A function is a special type of relation in which each element of the domain is paired with exactly one element in the range. A mapping shows how the elements are paired. It’s like a flow chart for a function, showing the input and output values. Maps present information about the world in a simple, visual way. They teach about the world by showing sizes and shapes of countries, locations of features, and distances between places. Maps can show distributions of things over Earth, such as settlement patterns.

Complete step-by-step answer:
We have $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}, \mathrm{f}(\mathrm{x})=\cos \mathrm{x}$
Let $\mathrm{f}\left(\mathrm{x}_{1}\right)=\mathrm{f}\left(\mathrm{x}_{2}\right)$
$\Rightarrow \cos x_{1}=\cos x_{2}$
$\Rightarrow \mathrm{x}_{1}=2 \mathrm{n} \pi \pm \mathrm{x}_{2}, \mathrm{n} \in \mathrm{Z}$
For example, when we use the function notation $f: R \rightarrow R,$ we mean that $f$ is a function from the real numbers to the real numbers. In other words, the domain of $\mathrm{f}$ is the set of real number $\mathrm{R}$ (and its set of possible outputs or codomain is also the set of real numbers $\mathbf{R}$ ).
Above equation has infinite solutions for $\mathrm{x}_{1}$ and $\mathrm{x}_{2}$.
Thus $\mathrm{f}(\mathrm{x})$ is many one function
Also the range of $\cos \mathrm{x}$ is [-1,1], which is a subset that is given a co-domain $\mathrm{R}$.
Hence function is not onto.
Hence, the correct answer is Option A.

Note: An example of mapping is creating a map to get to your house. An example of mapping is identifying which cell on one spreadsheet contains the same information as the cell on another spreadsheet. (mathematics) A function that maps every element of a given set to a unique element of another set; a correspondence. "Map" is a more general term than "translate", "rotate", etc.; it just means "transform every item in the domain" (and "domain" means "group of things we are transforming"). So, the "mapping notation" we have mentioned, like $(\mathrm{x}, \mathrm{y}) \rightarrow(\mathrm{x}+1, \mathrm{y}+1),$ is a way we can express any kind of transformation in the geometric plane.