Find the rules for the composite functions f dot function g and g dot function f.

f(x) = 9 square root of x + 7 ; g(x) = x^2 + 8

Is that 9\sqrt{x}+7 or 9\sqrt{x+7}?

Think of a composition of two functions as a process whereby one function first performs its "actions" on the variable, and then the result thereof is handed to the other function, which then performs its actions on the number it was given.

You can consolidate this multi-step process into a single function (the 'composite' function) by replacing the variable in the 'outside' function with the entire 'inside' function.

For example, using your f and g, the function f\ \circ \ g can be built by replacing f's variable with the g function, as...

f\ \circ \ g\ =\ 9\sqrt{(x^2+8)+7} . Note that in this function, the variable will first be squared, and 8 will be added to the result. That was g's job... now g is done and f takes over. 7 is added to the result; the number you have to that point is transformed to its positive square root; and then finally that result is multiplied by 9.

To build g\ \circ\ f replace g's variable with the function f.

Thanks for the catch, Galactus... I had the same question but then forgot to ask about it by the time I was finished.

It's probably 9\sqrt{x+7}, but I want to stress how important grouping symbols are. So many posters do not use them. Over time, we have gotten better at 'interpreting' what is actually meant.