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

Dec 3, 2017

Domain, $x \in \mathbb{R}$
Range, $f \left(x\right) \le 3$

#### Explanation:

First we can conisder the domain, this is fairly simple, we must consider what values of $x$ yields a valid value of $f \left(x\right)$, and we see for all values of $x$, $f \left(x\right)$ 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 $f \left(x\right)$ can take on, and by the sketch, we see the the max value of $f \left(x\right)$ is 3, this is the vertex point, where the vertex point is defined as being;
$\left(\frac{- b}{2 a} , f \left(\frac{- b}{2 a}\right)\right)$ as we can prove this rather simply using stationary points and differential calculus, and we see the vertex point is $\left(2 , 3\right)$

So from there $f \left(x\right)$ can take on any value lower than 3,
$\implies$ $f \left(x\right) \le 3$