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

Mar 5, 2017

Domain: $\left(- \infty , + \infty\right)$ Range: $\left(- \infty , 10\right]$

#### Explanation:

$y = 10 - {x}^{2}$

$y$ is defined $\forall x \in \mathbb{R}$

$\therefore$ the domain of $y$ is $\left(- \infty , + \infty\right)$

$y$ is a parabola with a maximum value where $y ' = 0$

$y ' = - 2 x = 0 \to x = 0$

${y}_{\max} = y \left(0\right) = 10$

$y$ has no finite lower limit.

Hence, the range of $y$ is $\left(- \infty , 10\right]$

This can be inferred by the graph of $y$ below:

graph{10-x^2 [-20.58, 19.95, -6.47, 13.79]}