1 4.1 Homework Part 5 of 8 Points: 0 of IF For $f(x)=x^{2}$ and $g(x)=x^{2}+1$, find the following composite functions and state the domain of each. (a) $f \circ g$ (b) $g \circ f$ (c) $f \circ f$ (d) $\mathrm{g} \circ \mathrm{g}$ (a) $(f \circ g)(x)=x^{4}+2 x^{2}+1 \quad$ (Simplify your answer.) Select the correct choice below and fill in any answer boxes within your choice. A. The domain of $f \circ g$ is $\{x \mid\}$. (Type an inequality. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. The domain of $f \circ g$ is all real numbers. (b) $(g \circ f)(x)=x^{4}+1 \quad($ Simplify your answer.) Select the correct choice below and fill in any answer boxes within your choice. A. The domain of $g \circ f$ is $\{x \mid$. (Type an inequality. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. The domain of $g \circ f$ is all real numbers. (c) $(f \circ f)(x)=\square($ Simplify your answer.)
Solution
Step 1 ：For the function \(f(x)=x^{2}\) and \(g(x)=x^{2}+1\), we need to find the following composite functions and state the domain of each.
Step 2 ：(a) The composite function \(f \circ g\) is calculated as \(f(g(x))\). Substituting \(g(x)\) into \(f(x)\), we get \(f(g(x)) = (x^{2}+1)^{2} = x^{4}+2x^{2}+1\). The domain of \(f \circ g\) is all real numbers.
Step 3 ：(b) The composite function \(g \circ f\) is calculated as \(g(f(x))\). Substituting \(f(x)\) into \(g(x)\), we get \(g(f(x)) = (x^{2})^{2}+1 = x^{4}+1\). The domain of \(g \circ f\) is all real numbers.
Step 4 ：(c) The composite function \(f \circ f\) is calculated as \(f(f(x))\). Substituting \(f(x)\) into \(f(x)\), we get \(f(f(x)) = (x^{2})^{2} = x^{4}\). The domain of \(f \circ f\) is all real numbers.
Step 5 ：Final Answer: \((f \circ f)(x) = \boxed{x^{4}}\) and the domain of \(f \circ f\) is \(\boxed{\text{all real numbers}}\).
