Consider f : R₊ → [4, ∞) given by f (x) = x² + 4. Show that f is invertible with inverse f ^{- 1} of given f by f ^{- 1} (y) = √ y - 4, where R₊ is the set of all non-negative real numbers

Solution:

The inverse of a function f(x) is a function g(x) such that if f maps an element ′a′ to an element ′b′, g maps ′b′ to ′a′.

According to the given problem,

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

For one-one:

Let f (x) = f (y)

⇒ x² + 4 = y² + 4

⇒ x² = y²

⇒ x = y [as x ∈ R]

Therefore,

f is a one -one function.

For onto:

For y ∈ [4, ∞), let y = x² + 4

⇒ x² = y - 4 ≥ 0 [as y ≥ 4]

⇒ x = √ (y - 4) ≥ 0

Therefore, for any y ∈ R, there exists x = √ (y - 4) ∈ R such that

f (x) = f (√(y - 4))

= √ (y - 4) ² + 4

= y - 4 + 4 = y

Therefore,

f is an onto function.

Thus, f is one-one and onto and therefore, f ^-1 exists.

Let us define g : [4, ∞) → R₊ by

g (y) = √ (y - 4)

Now, gof (x) = g (f (x))

= g (x² + 4) = √ (x² + 4 - 4)

= √ x² = x

And fog (y) = f (g (y))

= f (√ y - 4)= √ (y - 4) ² + 4

= (y - 4) + 4 = y

⇒ gof = fog = I_R

Hence, f is invertible and the inverse of f is given by

f ^-1 (y) = g (y) = √ (y - 4)

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.3 Question 8

Consider f : R₊ → [4, ∞) given by f (x) = x² + 4. Show that f is invertible with inverse f ^{- 1} of given f by f ^{- 1} (y) = √ (y - 4), where R₊ is the set of all non-negative real numbers

Summary:

Function f : R₊ → [4, ∞) given by f (x) = x² + 4 is invertible as gof = fog = I_R. The inverse of f is given by  f ^-1 (y) = g (y) = √ (y - 4)