Question

How do you find the domain and range of `f(x)= -2x^2+8x-5`?

Answer 1

Domain, `x in RR`
Range, `f(x) <= 3`

Explanation:

First we can conisder the domain, this is fairly simple, we must consider what values of `x` yields a valid value of `f(x)`, and we see for all values of `x`, `f(x)` is defined, and we can see that by a sketch; graph{-2x^2+8x-5 [-8.58, 11.42, -4.36, 5.64]}

To consider the range, we must cosnider all the values `f(x)` can take on, and by the sketch, we see the the max value of `f(x)` is 3, this is the vertex point, where the vertex point is defined as being;
`((-b)/(2a),f( (-b)/(2a) ) )` as we can prove this rather simply using stationary points and differential calculus, and we see the vertex point is `(2,3)`

So from there `f(x)` can take on any value lower than 3,
`=>` `f(x) <= 3`