Question

Given, what does this tell us about the relationship between the two functions?

Answer 1

one function is the inverse of the other

$\begin{gathered} f(x)=g^{-1}(x) \\ g(x)=f^-(x) \end{gathered}$

Explanation

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

Step 1

Function Composition is applying one function to the results of another. · (g º f)(x) = g(f(x)), first apply f(), then apply g()

so

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

and

Step 2

g(f(x))

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

n mathematics, an inverse is a function that serves to “undo” another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x,

so the realtion between the functios is

one function is the inverse of the other

g is the inverse of f

f is the inverse of g

$\begin{gathered} f(x)=g^{-1}(x) \\ g(x)=f^-(x) \end{gathered}$

I hope this helps you