If  is a relation on N, write the range of R.

If the function f: R  R  be given by  be given by  find fog and gof and hence find fog (2) and gof ( −3).

Discuss the commutativity and associativity of binary operation '*' defined on A = Q − {1} by the rule a * b = a − b + ab for all, a, b ∊ A. Also find the identity element of * in A and hence find the invertible elements of A. 

Consider f : R₊ → [−5, ∞), given by f(x) = 9x² + 6x − 5. Show that f is invertible with f⁻¹(y).

Hence Find
(i) f⁻¹(10)
(ii) y if f⁻¹(y)=43,

where R+ is the set of all non-negative real numbers.

If a*b denotes the larger of 'a' and 'b' and if a o b = (a*b) + 3, then write the value of (5) o (10), where and o are binary operations.

396.

Let A = { x ∈ Z: 0 ≤  x≤ 12} show that R = {(a,b):a,b ∈  A, |a-b|} is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also, write the equivalence class [2].

Show that the function f: R → R defined by$\mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{x}}{{\mathrm{x}}^2 + 1},\forall \mathrm{x}\in \mathrm{R}$ si neither one- one nor onto. Also, if g: R → R is defined as g(x) = 2x -1 find fog (x)

If f(x) = x + 7 and g(x) = x – 7, x $\in$ R, find (fog) (7)

(i) Is the binary operation *, defined on set N, given by  a * b = $\frac{\mathrm{a} + \mathrm{b}}{2}$  for all a,b $\in$N, commutative?

(ii) Is the above binary operation * associative?

If the binary operation * on the set of integers Z, is defined by a * b = a + 3b² , then find the value of 2 * 4.