Question

What is the domain and range for `f(x) = 3x - absx`?

Answer 1

Both the domain and the range are the whole of `RR`.

Explanation:

`f(x) = 3x-abs(x)` is well defined for any `x in RR`, so the domain of `f(x)` is `RR`.

If `x >= 0` then `abs(x) = x`, so `f(x) = 3x-x = 2x`.

As a result `f(x)->+oo` as `x->+oo`

If `x < 0` then `abs(x) = -x`, so `f(x) = 3x + x = 4x`.

As a result `f(x)->-oo` as `x->-oo`

Both `3x` and `abs(x)` are continuous, so their difference `f(x)` is continuous too.

So by the intermediate value theorem, `f(x)` takes all values between `-oo` and `+oo`.

We can define an inverse function for `f(x)` as follows:

`f^(-1)(y) = { (y/2, "if " y >= 0), (y/4, "if " y < 0) :}`

graph{3x-abs(x) [-5.55, 5.55, -2.774, 2.774]}