Question

1) Write the domain and range of Signum function.
2) A wheel makes 520 revolutions in 1 min. Through how many radians does it turn in 3 seconds.

Answer 1

Hint: We will first write the signum function and know about it in brief. After that we will look at the function and its domain and range. Now, for the other question, we will first find the revolutions in 1 sec and then multiply it by 3 and convert revolutions in angle in radians.

Complete step-by-step answer:
Part 1):
Let us get to know what a signum function is:-
\[f(x) = \dfrac{{|x|}}{x}\] is the signum function.
We can write is as:-
\[f(x) = \left\{ {\begin{array}{*{20}{c}}
  {1,x > 0} \\
  {0,x = 0} \\
  { - 1,x < 0}
\end{array}} \right.\]
The domain of a function is the set of all possible inputs for the function.
We can clearly see that we can put in any value of $x$ from Real Numbers.
Hence, the domain is $\mathbb{R}$.
Now, the set of all output values of a function is known as the range of a function.
By the definition of signum function, the range set is definitely {-1, 0, 1}.

Part 2):
We are given that a wheel makes 520 revolutions in 1 min.
We can write it as: Number of revolutions in 60 seconds = 520
Number of revolutions in 1 second will be = $\dfrac{{520}}{{60}} = \dfrac{{26}}{3}$.
Therefore, the number of revolutions in 3 sec will be:- $\dfrac{{26}}{3} \times 3 = 26$.
We know that 1 revolution means $2\pi $ radians of angle.
Hence, 26 revolutions will mean $2\pi \times 26 = 52\pi $.
Hence, the angle is $52\pi $ radians.

Note: The students may also take the angles in degree first and then convert them into radians but in the question, we were asked to answer the angle in radians. So, it is better to start off with radians only.
In mathematics, the sign function or signum function (from signum, Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as sgn.
There are special names for what can go into, and what can come out of a function:
What can go into a function is called the Domain.
What may possibly come out of a function is called the Codomain.
What actually comes out of a function is called the Range.