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

Jun 26, 2017

Domain: $\left(- \infty , + \infty\right)$
Range: $\left(- \infty , - 6\right]$

#### Explanation:

$f \left(x\right) - {x}^{2} + 4 x - 10$

$f \left(x\right)$ is defined $\forall x \in \mathbb{R}$

Hence the domain of $f \left(x\right)$ is $\left(- \infty , + \infty\right)$

$f \left(x\right)$ is a parabola of the form: $a {x}^{2} + b x + c$

Since the coefficient of ${x}^{2} < 0$, $f \left(x\right)$ will have a maximum value where $x = \frac{- b}{2 a}$

$\frac{- b}{2 a} = \frac{- 4}{- 2} = 2$

$\therefore {f}_{\text{max}} = f \left(2\right) = - {2}^{2} + 4 \cdot 2 - 10 = - 4 + 8 - 10 = - 6$#

Since $f \left(x\right)$ has no finite lower bound the range of $f \left(x\right)$ is $\left(- \infty , - 6\right]$

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