Question

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

Answer 1

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 `[0, a]` or `[–a, a]` where `a>0`, `R_f=[4-a^2,4]`
`D_f = [a, b]` where `a,b>0` and `b>a`, `R_f=[4-b^2,4-a^2]`
`D_f = [a, b]` where `a,b<0` and `b>a`, `R_f=[4-a^2,4-b^2]`
`D_f = [a, b]` where `a<0, b>0` and `abs(b)>abs(a)`, `R_f=[4-b^2,4]`
`D_f = [a, b]` where `a<0, b>0` and `abs(a)>abs(b)`, `R_f=[4-a^2,4]`

Ultimately however, since there is no restriction, the assumption is `D_f=RR`, thus `R_f=(–infty,4]`