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
(ii)             ƒ (x) =
1
x + 3
  ,   x ≠ -3
(iii)             ƒ (x) =
x - 1
x - 4
  ,   x ≠ 4
(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) =
1
x + 3
  ,   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 - 1
x - 4
  ,   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, ∞]

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