Question 127643
Let *[Tex m_1,m_2,b_1,b_2 \in R]


and 

{{{f(x)=m[1]x+b[1]}}} and {{{g(x)=m[2]x+b[2]}}}



So {{{(f o g)(x)=f(g(x))=m[1](m[2]x+b[2])+b[1]}}}



{{{m[1]m[2]x+m[1]b[2]+b[1]}}} Distribute



Since *[Tex m_1,m_2,b_1,b_2 \in R], this means that *[Tex m_1*m_2\in R] and *[Tex m_1*b_2\in R]. So this shows us that if f(x) and g(x) are linear, then (f o g)(x) is a linear function.





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Now let's verify this statement the other way.  




{{{(g o f)(x)=g(f(x))=m[2](m[1]x+b[1])+b[2]}}}



{{{m[2]m[1]x+m[2]b[1]+b[2]}}} Distribute



Since *[Tex m_1,m_2,b_1,b_2 \in R], this means that *[Tex m_2*m_1\in R] and *[Tex m_2*b_1\in R]. So this shows us that if f(x) and g(x) are linear, then (g o f)(x) is a linear function.