Composition of functions

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Question from Gilligan, a student:

Let f(x) = x^2, g(x) = 3x and h(x) = (sqrt{x}) + 1. Express each function as a composite of f, g and/or h.

(1) p(x) = 3(sqrt{x}) + 3

(2) Q(x) = (sqrt(sqrt{x}) + 1}) + 1

NOTE:

Question 2 is a radical within a radical.

So, sqrt{x} lies inside a BIGGER radical, which also includes + 1 in the radicand and OUTSIDE the radicand.

Is this clear?

Can someone answer number 1 with a clear explanation?

I should be able to answer number 2 following a reply to question 1.

Thanks

Gilligan,

The functions f, g and h do relatively straightforward things.

If you input the number x to f then f outputs the square of x.
If you input the number x to g then g outputs three times x.
If you input the number x to h then h outputs one more than the square root of x.

You can construct more complex functions however by taking the output of one function and making it the input to another function or even making it the input of the same function again. For example if you input x to h and take the output that h returns and input it to g then this is the composition g(h(x)). The result is

g(h(x)) = g(sqrt{x} + 1) = 3[sqrt{x} + 1] = 3(sqrt{x}) + 3

In the first step h outputs sqrt{x} + 1 and in the second step g outputs three times its input.

So what about g(g(x))?

g(g(x)) = g(3x) = 3(3x) = 9x

I hope this helps,
Penny

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