Consider f: R → R given by f(x) = 4x + 3. Show that f is in

1.	Consider f: R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
Answer» Given function f: R → R given by f(x) = 4x + 3 Let us show that the given function is invertible. Consider injection of f: Let x and y be any two elements of domain (R), Such that f(x) = f(y) ⇒ 4x + 3 = 4y + 3 ⇒ 4x = 4y ⇒ x = y Therefore, f is one-one. Now surjection of f: Let y be in the co-domain (R), Such that f(x) = y. ⇒ 4x + 3 = y ⇒ 4x = y - 3 ⇒ x = (y - 3)/4 in R (domain) ⇒ f is onto. Therefore, f is a bijection and, hence, is invertible. Let us find f^-1 Let f^-1(x) = y ...(1) ⇒ x = f(y) ⇒ x = 4y + 3 ⇒ x − 3 = 4y ⇒ y = (x - 3)/4 Now put these values in 1 we get Therefore, f^-1(x) = (x - 3)/4

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

Answer»

Given function f: R → R given by f(x) = 4x + 3

Let us show that the given function is invertible.

Consider injection of f:

Let x and y be any two elements of domain (R),

Such that f(x) = f(y)

⇒ 4x + 3 = 4y + 3

⇒ 4x = 4y

⇒ x = y

Therefore, f is one-one.

Now surjection of f:

Let y be in the co-domain (R),

Such that f(x) = y.

⇒ 4x + 3 = y

⇒ 4x = y - 3

⇒ x = (y - 3)/4 in R (domain)

⇒ f is onto.

Therefore, f is a bijection and, hence, is invertible.

Let us find f^-1

Let f^-1(x) = y ...(1)

⇒ x = f(y)

⇒ x = 4y + 3

⇒ x − 3 = 4y

⇒ y = (x - 3)/4

Now put these values in 1 we get

Therefore, f^-1(x) = (x - 3)/4