To study the simpler subclass obtained by composing function

To study the (simpler) subclass obtained by composing functions that send x to x + l or kx (for k not equalto 0). The latter functions obviously include any function of the form f(x) = ax + b with a notequalto 0, which is the result of multiplying by a. and then adding b. If f_1{x)=a_1x + b_1 with a_1 notequalto 0 and f_2(x) = a_2X+b_2 with a_2 notequalto 0, show that F_1 (f_2(x)) = Ax + B, with and find the constants A and B. Deduce from Exercise 5.5.1 that the result of composing any number of functions that send x to X + l or kx (for k notequalto 0) is a function of the form f(x) = ax + b with a notequalto 0. We know that such functions represent combinations of certain projections from lines to parallel lines, but do they include any projection from a line to a parallel line?

Solution

5.5.1

f1(f2(x))=a1f2(x)+b1=a1(a2x+b2)+b1=a1a2x+a1b2+b1=Ax+B

A=a1a2

B=a1b2+b1

5.52

Function that sends x to x+l is

f(x)=x+l and x to kx

is g(x)=kx

Both are linear functions as in 5.5.1 and their compositions will result in a function of the form Ax+B which is again linear and hence gives a linear function of the form ax+b when composed with another lienar function