Question

Answer the following:

Let f: R → R be a function defined by f(x) = 5x³ – 8 for all x ∈ R, show that f is one-one and onto. Hence find f ^–1 

Answer 1

f(x) = 5x³ – 8, x ∈ R

Let f(x₁) = f(x₂)

∴ 5x₁³ – 8 = 5x₂³ – 8

∴ x₁³ – x₂³ = 0

∴ `(x_1 - x_2) underbrace((x_1^2 + x_1 x_2 + x_2^2))_(>  0  "for all"  x_1 * x_2  "as discriminant"  <  0)` = 0

∴ x₁ = x₂

∴ f is a one-one function.

Let f(x) = 5x³ – 8 = y (say), y ∈ R

∴ x = `root(3)((y + 8)/5)`

∴ For every y ∈ R, there is some x ∈ R

∴ f is an onto function.

x = `root(3)((y + 8)/5)`

= f^–1 (y)

∴ f^–1 (x) = `root(3)((x + 8)/5)`