Q. 83.7( 25 Votes )
Consider f : R+→ [4, ∞) given by f (x) = x2 + 4. Show that f is invertible with the inverse f –1 of f given by , where R+ is the set of all non-negative real numbers.
It is given that f : R+→ [4, ∞) given by f (x) = x2 + 4.
Now, Let f(x) = f(y)
⇒ x2 + 4 = y2 + 4
⇒ x2 = y2
⇒ x = y
⇒ f is one-one function.
Now, for y ϵ [4, ∞), let y = x2 + 4.
⇒ x2 = y -4 ≥ 0
⇒ for any y ϵ R, there exists x = ϵ R such that
= y -4 + 4 = y.
⇒ f is onto function.
Therefore, f is one–one and onto function, so f-1 exists.
Now, let us define g: [4, ∞) → R+ by,
Now, gof(x) = g(f(x)) = g(x2 + 4) =
And, fog(y) = f(g(y)) = =
Therefore, gof = gof = IR.
Therefore, f is invertible and the inverse of f is given by
f-1(y) = g(y) =
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