For the real-valued functions $$ f(x)=x^{2}-3 $$ and $$ g(x)=x^{2}+3 $$, find the composition $$ f \circ g $$ and specify its domain using interval notation.

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The composition $$ f \circ g $$ is $$ x^4 + 6x^2 + 6 $$ and its domain is $$ (-\infty, \infty) $$. To find the composition $$ f \circ g $$, we need to evaluate $$ f(g(x)) $$. Given the functions: $$ f(x) = x^2 - 3 $$ $$ g(x) = x^2 + 3 $$ We start by substituting $$ g(x) $$ into $$ f(x) $$: $$ f(g(x)) = f(x^2 + 3) $$ Now, we replace $$ x $$ in $$ f(x) $$ with $$ g(x) = x^2 + 3 $$: $$ f(g(x)) = (x^2 + 3)^2 - 3 $$ Next, we need to expand $$ (x^2 + 3)^2 $$: $$ (x^2 + 3)^2 = x^4 + 6x^2 + 9 $$ Now, substituting this back into our expression for $$ f(g(x)) $$: $$ f(g(x)) = x^4 + 6x^2 + 9 - 3 $$ $$ f(g(x)) = x^4 + 6x^2 + 6 $$ Thus, the composition $$ f \circ g $$ is: $$ f \circ g = x^4 + 6x^2 + 6 $$ Next, we need to determine the domain of $$ f \circ g $$. The function $$ g(x) = x^2 + 3 $$ is defined for all real numbers $$ x $$, and since $$ f(x) $$ is also defined for all real numbers, the composition $$ f(g(x)) $$ is defined for all real numbers as well. Therefore, the domain of $$ f \circ g $$ is: $$ \text{Domain of } f \circ g = (-\infty, \infty) $$ In conclusion, the composition $$ f \circ g $$ is: $$ f \circ g = x^4 + 6x^2 + 6 $$ and its domain is: $$ (-\infty, \infty) $$

For the real-valued functions \( f(x)=x^{2}-3 \) and \( g(x)=x^{2}+3 \), find the composition \( f \circ g \) and specify its domain using interval notation.

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For the real-valued functions \( f(x)=x^{2}-3 \) and \( g(x)=x^{2}+3 \), find the composition \( f \circ g \) and specify its domain using interval notation.

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