Question

If the function f : R→ R be defined as `f(x) = (3x + 4)/(5x - 7), (x ≠ 7/5)` and g : R→ R be defined as `g(x) = (7x + 4)/(5x - 3), (x ≠ 3/5)` show that (gof)(x) = (fog)(x).

Answer 1

`g(f(x)) = (7f(x) + 4)/(5f(x) - 3)`   ...`(x ≠ 7/5, f(x) ≠ 3/5)`

= `(7((3x + 4)/(5x - 7)) + 4)/(5((3x + 4)/(5x - 7)) - 3)`

= `(21x + \cancel(28) + 20x - \cancel(28))/(\cancel(15x) + 20 - \cancel(15x) + 21)`

= `(41x)/41`

= x

`fog(x) = (3x + 4)/(5x - 7)`

= `(3g(x) + 4)/(5g(x) - 7)`   ...`(g(x) ≠ 7/5, x ≠ 3/5)`

= `(3((7x + 4)/(5x - 3)) + 4)/(5((7x + 4)/(5x - 3)) - 7)`

= `(21x + \cancel(12) + 20x - \cancel(12))/(\cancel(35x) + 20 - \cancel(35x) + 21)`

= `(41x)/41`

= x

gof(x) = fog(x)