Question

How do you find the domain and range of `-x^2 + 4x -10`?

Answer 1

Domain: `(-oo, +oo)`
Range: `(-oo, -6]`

Explanation:

`f(x) -x^2+4x-10`

`f(x)` is defined `forall x in RR`

Hence the domain of `f(x)` is `(-oo, +oo)`

`f(x)` is a parabola of the form: `ax^2+bx+c`

Since the coefficient of `x^2<0`, `f(x)` will have a maximum value where `x=(-b)/(2a)`

`(-b)/(2a) =(-4)/(-2) = 2`

`:. f_"max" = f(2) = -2^2+4*2-10 = -4+8-10 = -6`#

Since `f(x)` has no finite lower bound the range of `f(x)` is `(-oo, -6]`

We can see these from the graph of `f(x)` below.
graph{-x^2+4x-10 [-44.87, 37.33, -28.45, 12.64]}