Create explicit functions f: R-->R that satisfy the conditions?

Question

Condition 1:f is one-to-one and onto
2: f is one-to-one but not onto
3: f is onto but not one-to-one
4: f is neither one-one nor onto

Answer 1

1. If f is 1-1 and onto, every value of y must appear
in the range and no vertical line must hint the graph twice.
So y = x or y = x^3 work nicely.
2. Any function that takes only positive values and
is increasing will work. y = e^x will do fine.
3. Try y = x³-3x+1. This is onto, since it is a
cubic polynomial, but not 1-1 because it crosses
the x-axis 3 times.

4. y = sin(x) works, since the range of the
function is -1<= y <= 1.

Answer 2

1. f(x) = x, or f(x)=x^3 for a non-trivial example.
2. f(x) = e^x
3. f(x) = x^3 - 2x^2
4. f(x) = x^2