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{ = }(x)/(2+x) \\ g(x)\text{ = }(2x)/(1-x) \end{gathered}`

Step 3:

`\begin{gathered} f(g(x))\text{ = }\frac{(2x)/(1-x)}{2\text{ + }(2x)/(1-x)} \\ =\text{ }(2x)/(1-x)\text{ }(.)/(.)\text{ }\frac{2(1-x)\text{ +2x}}{1-x} \\ =\text{ }(2x)/(1-x)\text{ }(.)/(.)\text{ }\frac{2\text{ }}{1-x} \\ =\text{ }(2x)/(1-x)\text{ }*\text{ }(1-x)/(2) \\ =\text{ }(2x)/(2) \\ =\text{ }x \end{gathered}`

Next,

`\begin{gathered} g(f(x))\text{ = }\frac{2((x)/(2+x))}{1\text{ - }(x)/(2+x)} \\ =\text{ }(2x)/(2+x)\text{ }(.)/(.)\text{ }\frac{2\text{ + x - x}}{2+x} \\ =\text{ }(2x)/(2+x)\text{ }*(2+x)/(2) \\ =\text{ }(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