SOLUTION: 3. Show that the composition of two linear functions is also a linear function.

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Question 127643: 3. Show that the composition of two linear functions is also a linear function.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let

and
f%28x%29=m%5B1%5Dx%2Bb%5B1%5D and g%28x%29=m%5B2%5Dx%2Bb%5B2%5D


So %28f+o+g%29%28x%29=f%28g%28x%29%29=m%5B1%5D%28m%5B2%5Dx%2Bb%5B2%5D%29%2Bb%5B1%5D


m%5B1%5Dm%5B2%5Dx%2Bm%5B1%5Db%5B2%5D%2Bb%5B1%5D Distribute


Since , this means that and . 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.



%28g+o+f%29%28x%29=g%28f%28x%29%29=m%5B2%5D%28m%5B1%5Dx%2Bb%5B1%5D%29%2Bb%5B2%5D


m%5B2%5Dm%5B1%5Dx%2Bm%5B2%5Db%5B1%5D%2Bb%5B2%5D Distribute


Since , this means that and . So this shows us that if f(x) and g(x) are linear, then (g o f)(x) is a linear function.