How would you solve this composite function?

Question

Find both f(g(x)) and g(f(x)), given that f(x)=1/2x-2 and g(x)=2x+4.

Answer 1

Find both f(g(x)) and g(f(x)), given that f(x)=1/2x-2 and g(x)=2x+4.

** The fundamental idea behind composite function is that you replace the argument with a function, so f(g(x)) -> g(x) becomes "the x" in the argument / replacing everywhere you see x with g(x).

f(g(x)) = 1 / (2 * [2x + 4] - 2) [replace the x argument with the entire g(x) function]

= 1 / (4x + 8 - 2) [distribute the 2 into 2x + 4]

= 1 / (4x + 6) [simplify]

g(f(x)) = 2* [1 / (2x - 2)] + 4 [substitute f(x) everywhere there is a x in g(x)]

= [2 / (2x - 2)] + 4 [multiply 1 / (2x - 2) by 2]

= 2 / (2x - 2) + 4 * (2x-2) / (2x -2) [common denominator so you can add fractions]

= [2 + 4 * (2x - 2)] / (2x - 2) [add numerators and keep the denominator]

= [2 + 8x - 8] / (2x - 2) [simplify]

= (8x - 6) / (2x - 2) [simplify some more]

= 2/2 * (4x - 3) / (x - 1) [ pull 2 out of BOTH numerator and denominator]

= 1 * (4x - 3 ) / (x -1) = (4x - 3) / (x - 1)