Question

What is the domain and range of `f(x)=x^4-4x^3+4x^2+1`?

Answer 1

I will assume that since the variable is called `x`, we are restricting ourselves to `x in RR`. If so, `RR` is the domain, since `f(x)` is well defined for all `x in RR`.

The highest order term is that in `x^4`, ensuring that:

`f(x) -> +oo` as `x -> -oo`

and

`f(x)->+oo` as `x -> +oo`

The minimum value of `f(x)` will occur at one of the zeros of the derivative:

`d/(dx) f(x) = 4x^3-12x^2+8x`

`= 4x(x^2-3x+2)`

`= 4x(x-1)(x-2)`

...that is when `x = 0`, `x = 1` or `x = 2`.

Substituting these values of `x` into the formula for `f(x)`, we find:

`f(0) = 1`, `f(1) = 2` and `f(2) = 1`.

The quartic `f(x)` is a sort of "W" shape with minimum value `1`.

So the range is `{ y in RR: y >= 1 }`