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

Aug 29, 2015

Graph it

graph{4-x^2 [-10, 10, -5, 5]}

#### Explanation:

You'll notice that depending on your domains (denoted as ${D}_{f}$, you get different ranges (denoted as ${R}_{f}$):

D_f = [–a, 0] or $\left[0 , a\right]$ or [–a, a] where $a > 0$, ${R}_{f} = \left[4 - {a}^{2} , 4\right]$
${D}_{f} = \left[a , b\right]$ where $a , b > 0$ and $b > a$, ${R}_{f} = \left[4 - {b}^{2} , 4 - {a}^{2}\right]$
${D}_{f} = \left[a , b\right]$ where $a , b < 0$ and $b > a$, ${R}_{f} = \left[4 - {a}^{2} , 4 - {b}^{2}\right]$
${D}_{f} = \left[a , b\right]$ where $a < 0 , b > 0$ and $\left\mid b \right\mid > \left\mid a \right\mid$, ${R}_{f} = \left[4 - {b}^{2} , 4\right]$
${D}_{f} = \left[a , b\right]$ where $a < 0 , b > 0$ and $\left\mid a \right\mid > \left\mid b \right\mid$, ${R}_{f} = \left[4 - {a}^{2} , 4\right]$

Ultimately however, since there is no restriction, the assumption is ${D}_{f} = \mathbb{R}$, thus R_f=(–infty,4]