Property 15
Permutation Function
Let \(f:A\to A\) be a function. If f is a one to one and onto function, f is called a permutation function.
\(f\begin{pmatrix}a&b&c&d\\\downarrow&\downarrow&\downarrow&\downarrow\\b&d&a&c\end{pmatrix}\)
\(\Rightarrow f(a)=b\quad f(b)=d\quad f(c)=a\quad f(d)=c\)
1.
\(f\begin{pmatrix}a&b&c&d\\c&d&a&b\end{pmatrix}\)
\(\Rightarrow f(b)=?\)
2.
\(f\begin{pmatrix}1&2&3&4\\4&3&2&1\end{pmatrix}\)
\(\Rightarrow f(2)+f(3)=?\)
3.
\(f\begin{pmatrix}1&2&3&4\\4&2&1&3\end{pmatrix}\)
\(\Rightarrow f(2)-f^{-1}(3)=?\)
4.
\(f\begin{pmatrix}1&2&3&4&5\\5&3&2&1&4\end{pmatrix}\)
\(\Rightarrow f^{-1}(4)+f(1)=?\)
5.
\(f\begin{pmatrix}1&2&3&4\\4&3&1&2\end{pmatrix}\qquad g\begin{pmatrix}1&2&3&4\\1&3&4&2\end{pmatrix}\)
\(\Rightarrow (f\circ g)(3)=?\)
6.
\(f\begin{pmatrix}1&2&3&4\\3&1&2&4\end{pmatrix}\qquad g\begin{pmatrix}1&2&3&4\\2&1&3&4\end{pmatrix}\)
\(\Rightarrow (f\circ g^{-1})(1)=?\)
7.
\(f\begin{pmatrix}1&2&3&4\\4&2&1&3\end{pmatrix}\)
\(\Rightarrow (f\circ f)^{-1}(3)=?\)
Property 16
\(f:\mathbb{R}\to \mathbb{R}\text{ and }g:\mathbb{R}\to \mathbb{R}\)
\((f+g)(x)=f(x)+g(x)\)
\((f-g)(x)=f(x)-g(x)\)
\((f\cdot g)(x)=f(x)\cdot g(x)\)
\(\left(\samefrac{f}{g}\right)(x)=\samefrac{f(x)}{g(x)}\)
\((g(x)\ne0)\)
1.
\(f=\{(1,2)(2,4)(3,5)\}\)
\(\Rightarrow 2f(1)-3f(2)=?\)
2.
\(f=\{(2,3)(3,-4)(4,1)\}\)
\(g=\{(2,4)(3,1)(4,0)\}\)
\(\Rightarrow (f-g)(2)+(f\cdot g)(4)=?\)
3.
\(f=\{(1,-2)(2,5)\}\)
\(g=\{(1,-3)(2,2)\}\)
\(\Rightarrow (2f-g)(1)+(f\cdot g)(2)=?\)
4.
\(f(x)=3x+1\)
\(g(x)=4x-2\)
\(\Rightarrow (f+g)(2)+(2f+g)(1)=?\)
5.
\(f(x)=x+5\)
\(g(x)=2x-1\)
\(\Rightarrow (f+3g)(2)=?\)
6.
\(f(x)=3x+1\)
\(g(x-1)=2x\)
\(\Rightarrow (f+g)(1)=?\)
7.
\(f(x+1)=2x+1\)
\(g(x+1)=x-5\)
\(\Rightarrow (f\cdot g)(2)=?\)