Consider f : R → R given by f (x) = 4x + 3. Show that f is invertible. Find the inverse of f

Solution:

A function is a process or a relation that associates each element 'a' of a non-empty set A, to a single element 'b' of another non-empty set B.

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 → R given by f (x) = 4x + 3

For one-one

f (x) = f (y)

⇒ 4x + 3 = 4 y + 3

⇒ 4x = 4 y

⇒ x = y

Therefore,

f is a one-one function.

For onto

y ∈ R, let y = 4x + 3

⇒ x = (y - 3) ∈ R

Therefore, for any y ∈ R, there exists x = (y - 3)/4 ∈ R such that

f (x) = f [(y - 3)/4]

= 4 [(y - 3)/4] + 3 = y

⇒ f is onto.

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

Let us define g : R → R by

g (x) = (y - 3)/4

Now,

(gof )(x) = g (f (x)) = g (4x + 3)

= (4x + 3 - 3)/4 = x

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

= f [(y - 3)]/4

= 4 [(y - 3)/4] + 3

= y - 3 + 3 = y

⇒ gof = fog = I_R

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

f ^-1 (y) = g (y) = (y - 3)/4

---

NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.3 Question 7

Consider f : R → R given by f (x) = 4x + 3. Show that f is invertible. Find the inverse of f

Summary:

f: R → R is given by f (x) = 4x + 3 is invertible. The inverse of f is f ^-1 (y) = g (y) = (y - 3)/4