For f(x)=7x+8 and g(x)=x^{2}, find the following composite functions and state the domain of each. (a) f circ g (b) g circ f (c) f circ f (d) g circ g (a) (f circ g)(x)=7 x^{2}+8 (Simplify your answer.) Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The domain of f circ g is {x}. (Type an inequality. Use integers or fractions for any numbers in the expression. Use a comma to) B. The domain of f circ g is all real numbers. (b) (g circ f)(x)= (Simplify your answer.)

For the composite function \( (g \circ f)(x) = g(f(x)) \), we substitute \( f(x)=7x+8 \) into \( g(x)=x^{2} \). So we have: \[ (g \circ f)(x) = g(7x+8) = (7x+8)^{2} = 49x^2 + 112x + 64. \] As for the domain, since \( g(x) = x^2 \) is defined for all real numbers, the domain of \( g \circ f \) is also all real numbers. Now, let's express the domain of both functions clearly: A. The domain of \( f \circ g \) is all real numbers. B. The domain of \( g \circ f \) is also all real numbers. This means no matter what values of \( x \) we input, we can find corresponding outputs! For problem-solving tips, a common mistake when handling composite functions is to forget to correctly substitute the inner function before applying the outer function. Always check the order of operations! If composite functions get tricky, breaking them down stepwise can save your day!