Question

What is the domain and range of `y =4x - x^2`?

Answer 1

Domain: all `x in (-infty, infty)`, range: `y in (-infty,4]`

Explanation:

Domain is all `x`'s that the function `y` is not defined on, and in this case `y` is defined for all `x`'s.

To find the range notice you can factor `y` as `x(4-x)`. Therefore, the roots are at `0,4`. By symmetry you know that the maximum will take place in the middle of that, that will say when `x=2`. The reason its a max value is because of the negative sign on the `x^2` term, which will make the graph a "sad smiley".

So `max(y)=y(2)=4(2)-2^2=4`

As the functions greatest value is 4 and it goes to `-infty` as `x->+-infty` its range is all `y<=4`