Question

Find the domain and range of the real function
`f(x) = sqrt(9 -x^(2))`

Answer 1

If its clear that f(x) `= sqrt(9-x^(2))` is not defined when `(9-x^(2)) lt 0, i.e.,`
When `x^(2) gt 9 , i.e., " when " x gt 3 " or " x lt -3`
`:. " don " (f) ={ x in R : -3 le x le 3}`
Also `y= sqrt(9 - x^(2)) rArr y^(2) = (9 -x^(2))`
` rArr x= sqrt(9 -y^(2))`
Clearly x is not defined when `(9-y^(2)) lt 0`
But `(9-y^(2)) lt 0 rArr y^(2) gt 9`
`rArr y gt 3 " or " y lt -3`
`:.` range `(f) ={ y in R : - 3 le y le 3}`