The modulus, greatest–integer and signum functions are three standard real-valued…
MATHEMATICS · Multi-part · AoPS · Mathematics · Special Functions · Special Functions
Problem
The modulus, greatest–integer and signum functions are three standard real-valued functions that are frequently used in elementary mathematics.
(i) State the analytical definition of each of the following functions:
(a) f(x)=∣x∣ (b) g(x)=⌊x⌋ (c) h(x)=sgn(x).
(ii) On the same set of coordinate axes, sketch the graphs of y=f(x), y=g(x) and y=h(x) for −3≤x≤3.
(iii) Using your sketches, compare the domain and range of the three functions. Comment on the nature of any discontinuities you observe.
(iv) State, with reason, which of the three functions is an even function, which is an odd function, and which is neither even nor odd.