Question

Two functions are defined as: 

f(x)= (1/(-5x+19))+5 



g(x)= (1/(10x-3)) 

Find  fg(x) 
Simplify your answer into the form (ax+b)/(cx+d) where a,b,c,d, are numbers to be found. 

Please help, how is this answer found thankyou :)

Answer 1

multiply f(x) by g(x):
= ((1 / (-5x + 19)) + 5) * (1 / (10x - 3))
[first add the 5 to f(x)]
= ((1 + 5 * (-5x + 19)) / (-5x + 19)) * (1 / (10x - 3))
= ((1 -25x + 95) / (-5x + 19)) * (1 / (10x - 3))
= ((96 - 25x) / (-5x + 19)) * (1 / (10x - 3))
= (96 - 25x) / ((-5x + 19) * (10x -3))
= (96 - 25x) / (-50x^2 + 15x + 190x - 57)
= (25x + 96) / (50x^2 + 205x -57)

use the quadratic formula for the denominator: x = (-b ± √(b^2 - 4 *a * c)) / 2a
where for this equation a = 50, b = 205 and c = -57
write the answers in the form (x - answer one) (x - answer two)

then re-write the equation in the form (ax + b) / (cx + d)