Let f(x)=4x^2+x+1 and g(x)=x^2-2. find g(f( show each step of your work.

Hello there!

Today we are dealing with composite functions. I am going to introduce you to a new way of thinking about any function and this will help you in your understanding of composite functions.

So I want you to think of a function like a machine. Let's take a function and call it the f machine. If you take the input value 'x' and insert it into the f machine, you get f(x) as an output.

Now, for composite functions, the process is very similar BUT slightly more complicated. If we wanted to find g(f(x)), we would plug in 'x' to the f machine to get the output f(x). After that, we would plug the output into the g machine to get g(f(x)).

Let's try an example. Let's say f(x)=4x²+x+1 and g(x)=x²-2 and we want to find g(f(x)).
Well it looks like they already gave us the output for plugging x into the the f machine. The output is 4x²+x+1. Now we have to plug this into the g machine. Since the output that plugging x into the g machine is x²+2, and we want to find g(4x²+x+1), plug in 4x²+x+1 for x in the g machine. (This also means that g(4x²+x+1)=g(f(x)).)
Look...
g(4x²+x+1)=(4x²+x+1)²+2
which means...
g(f(x))=(4x²+x+1)²+2

Now simplify this...
(4x²+x+1)(4x²+x+1)+2
16x⁴+4x³+4x²+4x³+x²+x+4x²+x+1
16x⁴+8x³+9x²+2x+1

I really hope this helps!
Best wishes :)