(Recall that Z = {2, 1, 0, 1, 2, ...} is the integers, R the real numbers, (0,?) the positive
reals, and [0,?) the non-negative reals.)
Let $f: [0,\infty) \to [0, \infty); f(x) = x \cdot \lfloor x \rfloor$
Let $g: \mathbb{Z} \to \mathbb{Z}; g(x) = 2x + 3$
a) What is the domain and codomain of $f$?
b) What is the domain and codomain of $g$?
c) Is $f$ one-to-one? If not, show two elements of the domain which map to the same element
of the codomain.
d) Is $f$ onto? If not, show an element of the codomain which isnt mapped onto by any
element of the domain.
e) Is $g$ one-to-one? If not, show two elements of the domain which map to the same element
of the codomain.
f) Is $g$ onto? If not, show an element of the codomain which isnt mapped onto by any
element of the domain.
g) Is the inverse function $f^{-1}$ defined? If so, what is $f^{-1}(2)$? If not, why not?
h) Is the inverse function $g^{-1}$ defined? If so, what is $g^{-1}(2)$? If not, why not?
i) Is the composition $f \circ g$ defined? If so, what is $(f \circ g)(2)$? If not, why not?
j) Is the composition $g \circ f$ defined? If so, what is $(g \circ f)(2)$? If not, why not?
