Question

The range of a function is the set of all real numbers. Which of the following must also be true?
a) the domain of the function is the set of all real numbers
b) the function is a one to one function
c) the domain of the inverse of the function is the set of all real numbers
d) the range of the inverse of the function is the set of all real numbers

Answer 1

a) is false since the range of, for example,
\[\tan(x) \text{ for }-\frac{\pi}{2}<x<\frac{\pi}{2}\] is all real numbers

Answer 2

b) is false since for example
\[f(x)=x^3-3x\] has range all real numbers, but is not one to one
http://www.wolframalpha.com/input/?i=x^3-3x

Answer 3

c) doesn't make any sense because if the function is not one to one it does not have an inverse that is a function

Answer 4

d) is false because for example
\[\tan^{-1}(x)\] had range
\[(-\frac{\pi}{2},\frac{\pi}{2})\] and domain all real numbers