91Ó°ÊÓ

The function \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined as \(f(x)=\pi x-e\) is bijective. Find its inverse.

Consider the functions \(f, g: \mathbb{R} \rightarrow \mathbb{R}\) defined as \(f(x)=\sqrt[3]{x+1}\) and \(g(x)=x^{3}\). Find the formulas for \(g \circ f\) and \(f \circ g\).

Consider a function \(f: A \rightarrow B\) and a subset \(X \subseteq A\). We observed in Example 12.14 that \(f^{-1}(f(X)) \neq X\) in general. However \(X \subseteq f^{-1}(f(X))\) is always true. Prove this.

Consider a square whose side-length is one unit. Select any five points from inside this square. Prove that at least two of these points are within \(\frac{\sqrt{2}}{2}\) units of each other.

Give an example of a relation from \(\\{a, b, c, d\\}\) to \(\\{d, e\\}\) that is not a function.

A function \(f: \mathbb{Z} \rightarrow \mathbb{Z}\) is defined as \(f(n)=2 n+1\). Verify whether this function is injective and whether it is surjective.

A function \(f: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}\) is defined as \(f(m, n)=3 n-4 m\). Verify whether this function is injective and whether it is surjective.

Suppose \(f: \mathbb{Z} \rightarrow \mathbb{Z}\) is defined as \(f=\\{(x, 4 x+5): x \in \mathbb{Z}\\} .\) State the domain, codomain and range of \(f .\) Find \(f(10)\).

Consider the functions \(f, g: \mathbb{R} \rightarrow \mathbb{R}\) defined as \(f(x)=\frac{1}{x^{2}+1}\) and \(g(x)=3 x+2 .\) Find the formulas for \(g \circ f\) and \(f \circ g\).

Given a sphere \(S,\) a great circle of \(S\) is the intersection of \(S\) with a plane through its center. Every great circle divides \(S\) into two parts. A hemisphere is the union of the great circle and one of these two parts. Prove that if five points are placed arbitrarily on \(S,\) then there is a hemisphere that contains four of them.