# How do you find the domain and range of -x^2+7?

Dec 29, 2017

Domain: $x \in \mathbb{R}$
Range: $\left\{y | y \le 7\right\}$

#### Explanation:

The domain is rather simple to figure out. Since the function is defined with real numbers for all $x$, the domain will be $x \in \mathbb{R}$.

To figure out the range of the function, we'll consider it as a parabola. The leading term is negative so it'll be concave downwards ($\cap$). This means that the domain is $- \infty$ to whatever the vertex of the parabola is.

The vertex of the parabola $y = {x}^{2}$ is at $\left(0 , 0\right)$ and our function is that same parabola, but translated $7$ units up. This means the vertex will be at $\left(0 , 7\right)$ and this means our range is $- \infty$ to $7$:
$\left\{y | y \le 7\right\}$