How do you write #y = 2(x − 3)^5 − 5(x − 3)^2 + 0.5(x − 3) + 11# as a composition of two simpler functions?

1 Answer
Oct 27, 2015

#f(x)=2x^5-5x^2+0.5x+11#,

#g(x)=x-3#

Explanation:

There is no algorithm to solve this exercise, you only can see that your expression resembles a polynomial, since it is the sum of the powers of a certain quantity (namely #x-3#), with some coefficient. So, a polynomial described by words would be "pick a constant, add a certain quantity multiplied by some coefficient, then add the square of that quantity multiplied by some coefficient, and so on". And it is exactly what's happening here: you have #11#, then you have the first power of #x-3# multiplied by #0.5#, then you have the second power of #x-3# multiplied by #-5#, and finally you have the fifth power of #x-3# multiplied by #2#.

So, you have a function #f# which plays the role of the polynomial:

#f(x)=2x^5-5x^2+0.5x+11#,

and you're evaluating it at #g(x)=x-3# (what I called the "quantity" above).

So, the result is the combination #f(g(x))#