Show that f :[−1,1]→R,given by f(x)= is one-one. Find the inverse of the function f :[−1,1] Range f.
(Hint: For y ∈Range f, y = f(x)= , for some x in [−1, 1], i.e.,x=
f: [−1, 1] → R is given as f(x)=
Let f(x) = f(y).
xy+2x=xy+2y =>2x=2y
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=
=>xy+2y=x
x(1-y)=2y x= ,y≠1
Now, let us define g: Range f → [−1, 1] as g(y)= ,y≠1
Now,(gof)(x)=g(f(x))=g = .
Now,(gof)(x)=g(f(x))=g
∴gof = I[-1,1] and fog = Irangef
f−1 = g f-1 (y)= , y≠1
A certain reaction is 50 complete in 20 minutes at 300 K and the same reaction is 50 complete in 5 minutes at 350 K. Calculate the activation energy if it is a first order reaction. Given: