# What is the range of the function f(x)= -x^2 +9?

Sep 2, 2017

Range of $f \left(x\right) = \left[9 , - \infty\right)$

#### Explanation:

$f \left(x\right) = - {x}^{2} + 9$

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

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

Since the coefficient of ${x}^{2} < 0$ $f \left(x\right)$ has maximum value.

${f}_{\max} = f \left(0\right) = 9$

Also, $f \left(x\right)$ has no lower bounds.

Hence, the range of $f \left(x\right) = \left[9 , - \infty\right)$

We can see the range from the graph of $f \left(x\right)$ below.

graph{-x^2 +9 [-28.87, 28.87, -14.43, 14.45]}