How do you write y = 2(x − 3)^5 − 5(x − 3)^2 + 0.5(x − 3) + 11y=2(x3)55(x3)2+0.5(x3)+11 as a composition of two simpler functions?

1 Answer
Oct 27, 2015

f(x)=2x^5-5x^2+0.5x+11f(x)=2x55x2+0.5x+11,

g(x)=x-3g(x)=x3

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-3x3), 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 1111, then you have the first power of x-3x3 multiplied by 0.50.5, then you have the second power of x-3x3 multiplied by -55, and finally you have the fifth power of x-3x3 multiplied by 22.

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

f(x)=2x^5-5x^2+0.5x+11f(x)=2x55x2+0.5x+11,

and you're evaluating it at g(x)=x-3g(x)=x3 (what I called the "quantity" above).

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