Find the domain and range of the function f ( x ) = 1 x - 5 .

It is given that f ( x ) = 1 x - 5 We have to find the domain and range of f ( x ) Domain of f ( x ) is set of values x where f ( x ) is well-defined. For x = 5 , f ( x ) = 1 5 - 5 = 1 0 ⇒ f ( x ) is not well-defined at x = 5 Clearly that f ( x ) is real when x - 5 > 0 i.e., x > 5 Thus, the domain of the given function is ( 5 , ∞ ) . Range of f ( x ) is the set of values y such that f ( x ) = y Consider, f ( x ) = y = 1 x - 5 ⇒ x - 5 = 1 y ⇒ x - 5 = 1 y 2 ⇒ x = 1 y 2 + 5 For x ∈ ( 5 , ∞ ) , we have y ∈ R + . Hence, the range of f ( x ) is R + .

Find the domain and range of the function f ( x ) = 1 x - 5 .

It is given that f ( x ) = 1 x - 5 We have to find the domain and range of f ( x ) Domain of f ( x ) is set of values x where f ( x ) is well-defined. For x = 5 , f ( x ) = 1 5 - 5 = 1 0 ⇒ f ( x ) is not well-defined at x = 5 Clearly that f ( x ) is real when x - 5 > 0 i.e., x > 5 Thus, the domain of the given function is ( 5 , ∞ ) . Range of f ( x ) is the set of values y such that f ( x ) = y Consider, f ( x ) = y = 1 x - 5 ⇒ x - 5 = 1 y ⇒ x - 5 = 1 y 2 ⇒ x = 1 y 2 + 5 For x ∈ ( 5 , ∞ ) , we have y ∈ R + . Hence, the range of f ( x ) is R + .

If f ( x ) = y = ax - b cx - a , then prove that f ( y ) = x .

It is given that f ( x ) = y = ax - b cx - a We have to prove that f ( y ) = x Putting x = y in f ( x ) , we get f ( y ) = ay - b cy - a ⇒ f ( y ) = a ax - b cx - a - b c ax - b cx - a - a ⇒ f ( y ) = a ( ax - b ) - b ( cx - a ) cx - a c ( ax - b ) - a cx - a cx - a ⇒ f ( y ) = a 2 x - ab - bcx + ab cax - bc - cax + a 2 ⇒ f ( y ) = a 2 x - bcx a 2 - bc ⇒ f ( y ) = x a 2 - bc a 2 - bc = x Hence proved.