Question

Show that f :[−1,1]→R,given by f(x)= $\frac {x}{x+2}$ is one-one. Find the inverse of the function f :[−1,1] $\to$ Range f.
(Hint: For y ∈Range f, y = f(x)= $\frac {x}{x+2}$ , for some x in [−1, 1], i.e.,x= $\frac {2y}{1-y}$

Answer 1

f: [−1, 1] → R is given as f(x)= $\frac {x}{x+2}$
Let f(x) = f(y).
$\Rightarrow\frac {x}{x+2}=\frac{y}{y+2}$
$\Rightarrow$ xy+2x=xy+2y =>2x=2y
$\Rightarrow$ x=y
∴ f is a one-one function.
It is clear that f: [−1, 1] → Range f is onto.
∴ f: [−1, 1] → Range f is one-one and onto and therefore, the inverse of the function:
f: [−1, 1] → Range f exists.
Let g: Range f → [−1, 1] be the inverse of f.
Let y be an arbitrary element of range f.
Since f: [−1, 1] → Range f is onto, we have:y=f(x) for same x∈[-1,1]
y= $\frac {x}{x+2}$
=>xy+2y=x
x(1-y)=2y $\Rightarrow$ x=$\frac{2y}{1-y}$ ,y≠1
Now, let us define g: Range f → [−1, 1] as g(y)=$\frac{2y}{1-y}$ ,y≠1
Now,(gof)(x)=g(f(x))=g $(\frac {x}{x+2})$ = $\frac{2(\frac {x}{x+2})}{1-(\frac{x}{x+2})}=\frac{2x}{2}=x$.
Now,(gof)(x)=g(f(x))=g $(\frac{2y}{1-y})= \frac{\frac{2y}{1-y}}{\frac{2y}{1-y+2}}=\frac{2y}{2}=y.$
∴gof = I[-1,1] and fog = Irangef
$\therefore$ f−1 = g $\Rightarrow$ f-1 (y)=$\frac{2y}{1-y}$ , y≠1