Let F(x)=2x+4 and G(x)=4x^2 +4x

Question

After simplifying

(F o G) (x)=

When I did it on my paper I plugged in the 4x^2+4x in as F(4x^2+4x) and then fully plugged it in with it looking like 2(4x^2+4x)+4 and when I was multiplying I got 8x^2+8x+4 but I think I might be wrong and do not know where I went wrong if I got my work wrong on paper.

Answer 1

Your answer of 8x^2+8x+4 is correct. You could potentially simplify further by removing a greatest common factor of 4 to get 4(2x^2+2x+1)

If you are ever in doubt, you can always plug in a sample x value (such as x=0) into the original problem and your answer. If those numbers are equal, you have the correct equation. Sometimes, you are also able to factor out the squared function similar to when you solve a quadratic function, but this equation does not work for that.

Answer 2

Good news, your work is right. (F o G)(x) = 8x² + 8x + 4, which is exactly what you got, so you did not make a mistake anywhere.

Let me walk through it so you can see why, because being unsure about a correct answer usually means one step felt shaky even though you did it properly.

The notation (F o G)(x) means F(G(x)), and the way to read it is from the inside out. G is the inner function, so G acts on x first, and then whatever comes out of G gets fed into F. A lot of students read it left to right because that is how we read everything else, and that is the single most common error on these.

So we start by writing F with G inside it:

(F o G)(x) = F(G(x)) = F(4x² + 4x)

Now F is the rule "double it and add 4." It does not care what you hand it. If you hand it x it gives 2x + 4, and if you hand it 4x² + 4x then it doubles that whole thing and adds 4:

2(4x² + 4x) + 4

Distributing the 2 across both terms inside the parentheses:

8x² + 8x + 4

which is your answer. The parentheses were the important part, since you have to double the entire inside, not just the first term, and you kept them, which is why it came out right.

If you notice, there is nothing left to combine here. 8x² and 8x are not like terms because one has x squared and the other has x to the first power, and 4 is just a constant, so the expression is already fully simplified. I think that may be what made you second guess yourself, since usually there is a last step where things collapse together and here there is not.

Remember, our purpose is F(G(x)) and not G(F(x)), and those two give completely different answers. If you had gone the other direction, you would be plugging 2x + 4 into G, which looks like 4(2x + 4)² + 4(2x + 4), and that works out to 16x² + 72x + 80. So the order really matters, and that is worth noticing because composition is one of the few operations in algebra where swapping the two things changes the result.

One easy way to check yourself on any composition problem is to test a number. Take x = 1. Going step by step, G(1) = 4(1) + 4(1) = 8, and then F(8) = 2(8) + 4 = 20. Now test your formula at the same value, 8(1) + 8(1) + 4 = 20. They match, so the formula is right. I also tried x = 2, where G(2) = 24 and F(24) = 52, and the formula gives 32 + 16 + 4 = 52. When two different inputs both match, you can be confident.

So, our final answer is 8x² + 8x + 4, and your paper was correct.