Let f : \(R\rightarrow R\)  be defined as f(x) = 10x + 7. Find the function g : \(R\rightarrow R\) such that gof = fog = I_R

Let A = R- 5(3), B = R-[1] Let \(f:A\rightarrow B\) be defined by \(f(x)=\left( \frac { x-2 }{ x-3 } \right) \forall x\in A\). Then show that f is bijective Hence find  \(f^{ -1 }(x)\)

Let x be a non-empty set and * be a binary operation on P(X) (the power set of set X) defined by
\(A*B=A\cup B\ for\ all\ A,B\in P(x)\)
Prove that '*' is both commutative and associative on P(X). Find the identity element with respect to on P(X). Also, show that <1>E P(X) is the only invertible element of P(X).

Check the injectivity and surjectivity of the following
(i) f : N \(\rightarrow\) N given by f(x) = x²
(ii) f : R \(\rightarrow\) R given by f(x) = x²
(iii) f : Z → Z given by f(x) = x²
(iv) f : N → N given by f(x) = x³
(v) f : Z → Z given by f(x) = x³

Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let R₁ be a relation in X given by R₁ = {(x, y)}: x - y is divisible by 3} and R₂ be another relation on X given by R₂ = {(x, y): (x, y)} {1, 4, 7} or {x, y} {2, 5, 8} or {x, y} {3, 6, 9}} show that R₁ = R₂ 

Consider \(f:R_{ + }\rightarrow [-5,\infty )\) given by f(x) = 9x² + 6x - 5 Show that f is invertible find f^-1(x) where R₊ is the set of all non-negative real numbers.

Let A = \(R\times R\) and * be a binary operation on A defined by 
(a,b)*(c,d) = (a+c, b+c)
Show that * is commutative and associative. Find the identity element for * on A. Also find • the inverse of every element (a, b)  \(\in A\) *.