Question

find g(f(x)) when f(x)=x^2+1 g(x)= sqrt(x+3)

Answer 1

First thing you have to do is find what f(x) is, and then apply that function to g(x). That would be done like this: If f(x) is x^2 + 1, you fill x^2 + 1 into the "x" in the g(x) function. \[g(f(x))=\sqrt{(x ^{2}+1)+3}\]That would simplify down to x^2 + 4. Do you understand how to do the composite thing?

Answer 2

Isn't there another x so it would be cubed? would it be \[\sqrt{x^3+4}\]

Answer 3

No. You are just replacing the "x" in the g function with x^2 + 1. That's it.

Answer 4

Do you want a simpler-looking example? We could do that!

Answer 5

If f(x) has x^2 and you are multiplying that with g's x wouldn't it turn into x^3? I'm confused where one of the 3 x's is being dropped

Answer 6

@joeyanddior Each function's 'x' variable is a different variable. Try think of it as \(f(x)=x^2 + 1\) and \(g(y)=\sqrt{y+3}\)
So when you go \(g(f(x))\) that's basically saying Let \(y=f(x)=x^2+1\), so \(g(y)=g(x^2+1)\) <- can you see from this that you need to "replace" the 'y' in the g(y) equation with what you are putting into the function?

Answer 7

Ok, let's look at a simpler example. Let's say that f(x) = x + 1 and g(x) = x + 2. Find g(f(x)). Well, f(x) = x + 1. When you apply f's function rule to g(x), all you do is replace the x in g(x) with the function rule in f(x). Since f(x) = x + 1, every x in g is now x + 1. So the composite g(f(x)) would be (x + 1) + 2. See? That "x" in the g function now has the whole f function as its replacement. Like, even easier, let's say that f(x) = x + 1 and g(x) = x. If we need to find g(f(x)), we take f's function, x + 1 and put it into g's function. since g(x) = x, applying f's rule to it would give you x + 1.

Answer 8

Where did the composite thing come from?