Find the domain and range of the following real function:

(i) f(x) = –|x|, x ∈ R 

We know that

 || = { , ≥ 0;   −, < 0 

∴ () = −|| = { −, ≥ 0 ; , < 0 

Since f(x) is defined for x ∈ R, the domain of f is R. 

It can be observed that the range of

 f(x) = –|x| is all real numbers except positive real numbers. 

∴ The range of f is (−∞, 0]. 

(ii) () = √(9 − ²) 

Since √(9 − ²) is defined for all real numbers that are greater than or equal to –3 and less than or equal to 3, the domain of f(x) is {x : –3 ≤ x ≤ 3} or [–3, 3]. For any value of x such that –3 ≤ x ≤ 3, the value of f(x) will lie between 0 and 3. 

∴The range of f(x) is {x: 0 ≤ x ≤ 3} or [0, 3].