Question

Given f(x)=\frac{x}{2+x} and g(x)=\frac{2x}{1-x} find f(g(x)) and g(f(x)). What does this tell us about the relationship between the two functions? To earn full credit show all work/calculations in finding the compositions. You may want to write this work out by hand and upload a picture of that hand written work rather than trying to type it all out.

Answer 1

Step 1:

The inverse composition rule.

The functions f(x) and g(x) are inverses when f(g(x)) = f(g(x)).

Step 2:

$\begin{gathered} f(x)\text{ = }\frac{x}{2+x} \\ g(x)\text{ = }\frac{2x}{1-x} \end{gathered}$

Step 3:

$\begin{gathered} f(g(x))\text{ = }\frac{\frac{2x}{1-x}}{2\text{ + }\frac{2x}{1-x}} \\ =\text{ }\frac{2x}{1-x}\text{ }\frac{.}{.}\text{ }\frac{2(1-x)\text{ +2x}}{1-x} \\ =\text{ }\frac{2x}{1-x}\text{ }\frac{.}{.}\text{ }\frac{2\text{ }}{1-x} \\ =\text{ }\frac{2x}{1-x}\text{ }\times\text{ }\frac{1-x}{2} \\ =\text{ }\frac{2x}{2} \\ =\text{ }x \end{gathered}$

Next,

$\begin{gathered} g(f(x))\text{ = }\frac{2(\frac{x}{2+x})}{1\text{ - }\frac{x}{2+x}} \\ =\text{ }\frac{2x}{2+x}\text{ }\frac{.}{.}\text{ }\frac{2\text{ + x - x}}{2+x} \\ =\text{ }\frac{2x}{2+x}\text{ }\times\frac{2+x}{2} \\ =\text{ }\frac{2x}{2} \\ =\text{ x} \end{gathered}$

Final answer

So we see that functions f(g(x)) and g(f(x)) are inverses because f(g(x)) = f(g(x)) = x