Is there an efficient method to find the range of compositions of functions?
I know this: The domain of the composition of functions is the INTERSECTION of the domain of the INSIDE function and the domain of the RESULTING function.
However, I'm struggling to find the range of compositions of functions. Is the range the INTERSECTION of the range of the OUTSIDE function and the range of the resulting function? I don't think this assumption is correct though. Can someone please help?
For example, let's define two functions: $f(x)= \sqrt{x}$ and $g(x)= x+1$. The resulting function, $f(g(x))$ is $\sqrt{x+1}$. This resulting function has a domain of greater than or equal to $-1$, and it has a range of greater than or equal to $0$.
In the example, $f(x)$ has a domain of greater than or equal to $0$, $g(x)$ has a domain of all real numbers, $f(x)$ has a range of greater than or equal to $0$, $g(x)$ has a range of all real numbers.
The domain of the composition is the INTERSECTION of the domain of the INSIDE function (in this case, $g(x)$ has a domain of all real numbers), and the domain of the RESULTING function (in this case, $f(g(x))$, or $\sqrt{x+1}$, has a domain of greater than or equal to $-1$). Hence, the final domain of the composition is GREATER THAN OR EQUAL TO $-1$.
What is the range??
Thank you!
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