Consider \(f:R_+\rightarrow[-5,\infty) \) given by f(x) = 9x² + 6x - 5. Show that f is invertible with \(f^{-1}(y)={(\sqrt{(y+6)}-1)\over3}\)

Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a*b = min{a, b}. Write the operation table of the operation *.

Find \(fof^{ -1 }\) and \(f^{ -1 }\) of for the function:
\(f(x)=\frac { 1 }{ x } ,x\neq 0\). Also prove that \(fof^{ -1 }\)= \(f^{ -1 }\) of .

Let R be a relation defined on the set of natural numbers Nas follow:
R = {(x, y) : x \(\in \) N, y \(\in \)N and 2x + y = 24}
Find the domain and range of the relation R. Also, find if R is an equivalence relation or not.

Let X be a non-empty set, Let * be a binary operation on the power set P(X) defined by A * B = A n B. What is the identify element for the operation * ? Given X is a set of people in a locality, A is a set of children and B is a set of citizens aged above 75 years in the same locality. Is * an invertible binary Feration for these sets as defined above?
What qualities would you suggest that elements of A should have towards elements of B?

Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as \(a * b=\left\{\begin{array}{ll} a+b, & \text { if } a+b<6 \\ a+b-6 & \text { if } a+b \geq 6 \end{array}\right.\) show that 0 is the identity for this operation and each element of the set is invertible with 6 - a being the inverse of  a.