Question

What are examples of a function which is (a) onto but not one-to-one; (b) one-to-one but not onto, with a domain and range of `(-1,+1)`?

Answer 1

Examples
onto but not one-to-one: `f(x)=1/(x+2)`
one-to-one but not onto: `g(x)=2abs(x)-1`

Explanation:

If `f(x)=1/(x+2)`
then
`color(white)("XXX")`if `f(x_1)=f(x_2)`
`color(white)("XXX")`then (noting that for `x_1, x_2 in (-1,+1)`)
`color(white)("XXXXXXX")x_2+2=x_1+2` (after cross multiplying)
`color(white)("XXXXXXX")rarr x_2=x_1`
`color(white)("XXXXXXX")`which implies `f(x)` one-to-one

however
`color(white)("XXX")`there is no value `barx` for which
`color(white)("XXX")f(barx)=0` (which is a value in the specified Range: `(-1,+1)`)
`color(white)("XXX")`which implies `f(x)` is not onto.

If `g(x)=2abs(x)-1`
then
`color(white)("XXX")`any value `g(x) in (-1,+1)`
`color(white)("XXX")`can be generated by some value of `x`
`color(white)("XXX")`which implies `g(x)` is onto
however
`color(white)("XXX")`if `x_1=-1/2` and `x_2=+1/2`
`color(white)("XXX")`then `g(x_1)=g(x_2)` but `x_1!=x_2`
`color(white)("XXX")`which implies `g(x)` is not one-to-one