Question

A function f: R → R is defined as f(x) = x³ + 4. Is it a bijection or not? In case it is a bijection, find f⁻¹(3).

Answer 1

Given function f: R → R is defined as f(x) = x³ + 4

Injectivity of f:

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

Such that f(x) = f(y)

⇒ x³ + 4 = y³ + 4

⇒ x³ = y³

⇒ x = y

Therefore, f is one-one.

Surjectivity of f:

Let y be in the co-domain (R),

Such that f(x) = y.

⇒ x² + 4 = y 

⇒ x³ = y – 4

⇒ x = ³√(y – 4) in R (domain)

⇒ f is onto.

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

Find f^-1

Let f⁻¹(x) = y …(1)

⇒ x = f(y)

⇒ x = y³ + 4

⇒ x − 4 = y³

⇒ y = ³√(x - 4)

Therefore, f^-1(x) = ³√(x - 4) [from (1)]

f^-1 (3) = ³√(3 – 4)

= ³√-1

= -1