Find the domain of combined functions

Question

I have a question asking to find the domain of $g(f(x))$ given $f(x)=2x^2+x$, and $g(x)=x^2+1$. I can easily do these questions in reverse when you have to find $f(g(x))$, but when having to find $g(f(x))$ I get a little mixed up. Here is how I started:

$g(f(x))$

$=x(2x^2+x)^2+1$

$=2x^2+x(2x^2+x)+1$

But I am not sure if I have set this up properly.

If someone could help me through this, I'd really appreciate it.

Answer 1

You have the right idea, but you just have to do one thing at a time. Here,

$g(f(x))$

$= g(2x^2 + x)$

$= (2x^2 + x)^2 + 1$

Just think of $g$ as a machine that takes its input, and outputs the input squared plus one.

Now, assuming that the domains of $f$ and $g$ are both $\mathbb{R}$, see if there are any real numbers $x$ for which the above expression creates a problem. If there are, those values are not part of the domain. If there are no such values, then the domain is all of $\mathbb{R}$.

Answer 2

$g(f(x)) = g(2x^2+x) = (2x^2+x)^2+1 = (2x^2)^2 + 2(2x^2)x + x^2 + 1 = ....$ can you take it from here?

Answer 3

Here is a simple rule: the domain of all polynomial functions(a sum of positive powers of x) is all Real numbers unless you are told otherwise. Additionally the below also holds for polynomials:

1. They are continuous on R
2. They are differentiable on R

So when you see functions such as below, rejoice!

f(x)=-2x^2+x-7 f(x)=x^3/3-3x^2+0.755 f(x)=4x^4+3x-55