Class 12 Mathematics - Chapter Relations and Functions NCERT Solutions | Show that f : [–1, 1] → R, gi
Show that f : [–1, 1] → R, given by
is one-one. Find the inverse of the function f : [–1, 1] → Range f.
(Hint: For y ∈ Range f, y =, for some x in [ - 1, 1], i.e.,
)
f: [ - 1, 1] → R is given as
Let f(x) = f(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:
Now, let us define g: Range f → [ - 1, 1] as
∴gof =I[-1, 1]and fog = IRange f
∴ f - 1 = g
⇒ 
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