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Question -

Consider f: R → R+ → [4, ∞) givenby f(x) = x2 + 4. Show that f is invertiblewith inverse f−1 of f given by f−1(x)= √ (x-4) where R+ is the set of all non-negative real numbers.



Answer -

Givenf: R → R+ → [4, ∞) given by f(x) = x2 +4.

Now we have to showthat f is invertible,

Consider injectionof f:

Let x and y be two elements of the domain (Q),

Such that f(x) =f(y) 

⇒ x2 +4 = y2 + 4

⇒ x2 =y2

⇒ x = y     (as co-domain as R+)

So, f isone-one

Now surjectionof f:

Let y be in the co-domain (Q),

Such that f(x) =y

⇒ x2 +4 = y

⇒ x2 =y – 4

⇒ x = √ (y-4) in R

⇒ f is onto.

So, f is a bijection and, hence, it is invertible.

Now we have tofind f-1:

Let f−1 (x) = y……(1)

⇒ x = f (y)

⇒ x = y2 + 4

⇒ x − 4 = y2

⇒ y = √ (x-4)

So, f-1(x)= √ (x-4)

Now substituting thisvalue in (1) we get,

So, f-1(x)= √ (x-4)

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