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Sample Problems. Decompose each of the following functions as a composite.
  1. H(x) = (2x+1)3
  2. J(x) = 2x3 + 1
  3. K(x) = sqrt(6x-5)
  4. L(x) = sin(7x2-4)

Solutions. We can build the functions H and J from these two simpler functions.

f(x) = 2x+1 and g(x) = x3.

With these simpler functions, we find that
H(x) = g(f(x)): (first multiply by two and add one, then cube) and

J(x) = f(g(x)): (first cube, then multiply by two and add one).

In order to decompose K, we note that when K is applied to a real number x, it first multiplies x by 6 and subtracts 5; the last (or outermost) step is to take the square root. We therefore take m to be the linear function given by m(x)=6x-5. We find that

K(x) = sqrt(m(x)).

In order to decompose L, we form the quadratic function q(x)=7x2 - 4 and we note that

L(x)=sin(q(x))

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