Find the domain and range of the following real functions:(i) `f(x)=-|x|` (ii) `f(x)=sqrt(9-x^2)`

Answer

N/a

Explanation

(i) `f(x)=-|x|`
f (x), is defined of all real values of x.
`therefore"Domain"=R`
`Let y=-|x|impliesyge0`
`implies,y in]-oo","0]`
`therefore"Range of"f(x)=]-oo,0]`
(ii) `f(x)=sqrt(9-x^(2))`
Domain `- f (x)` is defined if `9-x^(2)ge0`
`implies(3-x)(3+x) ge0`
`implies3-x ge0,3 +x ge0`
or `3-xle0,3+xle0`
`impliesxle3,xge-3or xge3,xle-3`
Now `xge3,xle-3` is impossible.
`xle3,xge-3 implies-3lexle3 implies"Domin" of f(x)=[-3,3]`
Range, `"let "y=sqrt(9-x^(2))impliesyge0`
`impliesy^(2)=9-x^(2)impliesx^(2)=9-y^(2)`
`impliesx=sqrt(9-y^(2))impliesy in[-3,3]`
Now, `yge 0 and y in[-3,3]`
`impliesy in[0,3]`
Therefore, range of f (x) = [0,3]