# Q 8 ) . Consider f : R+ → [4, ∞) given by . Show that f is invertible with the inverse  of f given by   , where R+ is the set of all non-negative real numbers.

It is given that
,    and

Now, Let f(x) = f(y)

⇒ x2 + 4 = y2 + 4

⇒ x2 = y2

⇒ x = y

⇒ f is one-one function.

Now, for y ? [4, ∞), let y = x2 + 4.

⇒ x2 = y -4 ≥ 0 ⇒ for any y ? R, there exists x = ? 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,

g(y) = 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) = Rate this question :

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