Fsc Part 2 Mathematics (Complete Solutions)
Q3
Without finding the inverse, state that domain and range of...
Without finding the inverse, state that domain and range of ƒ^{1}.
(i) ƒ (x) = √x + 2




(iv) ƒ (x) = (x  5)^{2} , x ≥ 5

Solution
(i) ƒ (x) = √x + 2
Here ƒ is not defined when x < 2
Domain of ƒ = [2, ∞]
By definition of inverse of function
Domain of ƒ = Range of ƒ^{1}
So for each ƒ^{1} (x) ∈ [2, ∞] there correspond the domain set [0, ∞] of ƒ^{1}
Hence Domain of ƒ^{1} = [0, ∞]
and Range of ƒ^{1} = [2, ∞]
(ii) ƒ (x) = 

, x ≠ 3 
Here ƒ is not defined when x = 4
Domain of ƒ = Range  {4} = Range of ƒ^{1}
So for each ƒ^{1} (x) ∈ R  {4} we have
Domain set of ƒ^{1} = R  {1}
Hence Domain of ƒ^{1} = R  {1}
and Range of ƒ^{1} = R  {4}
(iii) ƒ (x) = 

, x ≠ 4 
Here ƒ is not defined when x = 3
Domain of ƒ = Range  {3} = Range of ƒ^{1}
So for each ƒ^{1} (x) ∈ R  {3} we have the
Domain set of ƒ^{1} = R  {0}
Hence Domain of ƒ^{1} = R  {0}
and Range of ƒ^{1} = R  {3}
(iv) ƒ (x) = (x  5)^{2} , x ≥ 5
For x ≥ 5 we have corresponding range the set of positive real number including "0"
Domain of ƒ = [5, ∞]
Range of ƒ = [0, ∞]
By the definition of inverse of function
Hence Domain of ƒ^{1} = [0, ∞]
and Range of ƒ^{1} = [5, ∞]