Let f: R → R and g: R → R be defined by f(x) = x2 and g(x) = x + 1. Show that fog ≠ gof.

Let A be the set of first five natural and let R be a relation on A defined as follows: (x, y) R x ≤ y

Express R and R^-1 as sets of ordered pairs. Determine also

(i) the domain of R^‑1

(ii) The Range of R.

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).

If f: R → R be defined by f(x) = x³ −3, then prove that f⁻¹ exists and find a formula for f⁻¹. Hence, find f⁻¹ (24) and f⁻¹ (5).

Consider f: R⁺ → [−5, ∞) given by f(x) = 9x² + 6x − 5. Show that f is invertible with f^-1(x) = \( \dfrac{\sqrt{(x+6)-1}}{3}\)

If f(x) = \(\dfrac{ (4x + 3)}{(6x – 4)}\), x ≠ (\( \dfrac{2}{3}\)) show that fof(x) = x, for all x ≠ (\( \dfrac{2}{3}\)). What is the inverse of f?