Suppose that the functions f and g are defined as followsSol

Suppose that the functions f and g are defined as follows.

Solution

We have f(x) = 1/( 3x²+5) and g(x) = 1/ (5x-1) . Then f.g(x) = f (g (x) ) = f ( 1/(5x-1) ) = 1/[ 3 {(1/(5x-1)}² +5] = 1/ [ 3/( 5x-1) + 5} = 1/ [ ( 3 + 25x -5)/(5x -1) = (5x-1)/ (25x -2).

Also ( f + g )(x) = f(x) +g(x) = [1/(3x² +5)] + [1/( 5x-1)]

As regards the domain of ( f. g) (x) = (5x-1)/ (25x -2), we may observe that x can take all real values except – 2/25 , as (25x -2) = 0 when x = -2/25. Since division by 0 is not defined, hence ( f . g ) (x) is indeterminate when x = -2/25. Thus, the domain of (f . g )(x) is (- , 2/25) U ( 2/25, ).

As regards the domain of ( f. g) (x) = [1/(3x² +5)] + [1/( 5x-1)] , since (5x-1) is real only when 5x > 1 or when x > 1/5 ( x cannot be equal to 1/5 as this will make ( 5x-1) = 0 and division by 0 is not defined; also x cannot be less that 1/5 as this will make ( 5x-1) imaginary).Thus, the domain of (f + g ) (x) is ( 1/5, ).