Let's examine the natural domains of each function in parts (a) and (b) to determine where each function is defined.

Part (a)

Given: $f(x) = \frac{x^2 + x}{x + 1} \quad \text{and} \quad g(x) = x$

1. Domain of $f(x)$:

   • The function $f(x)$ has a denominator $x + 1$, which means it is undefined when $x + 1 = 0$.
   • Solving $x + 1 = 0$, we get $x = -1$.
   • Therefore, the domain of $f(x)$ is all real numbers except $x = -1$: $\mathbb{R} \setminus \{-1\}$.

2. Domain of $g(x)$:

   • The function $g(x) = x$ is a polynomial, which is defined for all real values of $x$.
   • So, the domain of $g(x)$ is $\mathbb{R}$.

Comparison:

• The domain of $f(x)$ is $\mathbb{R} \setminus \{-1\}$.
• The domain of $g(x)$ is $\mathbb{R}$.
• Therefore, the domain of $g(x)$ is slightly larger than that of $f(x)$ due to the restriction at $x = -1$ in $f(x)$.

Part (b)

Given: $f(x) = \frac{x \sqrt{x} + \sqrt{x}}{x + 1} \quad \text{and} \quad g(x) = \sqrt{x}$

1. Domain of $f(x)$:

   • The square roots $\sqrt{x}$ in both the numerator and denominator imply that $x$ must be non-negative (i.e., $x \geq 0$) for the square roots to be defined.
   • Additionally, the denominator $x + 1$ cannot be zero, so $x \neq -1$. However, since $x \geq 0$ already excludes $x = -1$, this constraint does not affect the domain further.
   • Thus, the domain of $f(x)$ is $x \geq 0$, or $[0, \infty)$.

2. Domain of $g(x)$:

   • The function $g(x) = \sqrt{x}$ also requires $x \geq 0$ for the square root to be defined.
   • So, the domain of $g(x)$ is $[0, \infty)$.

Comparison:

• Both $f(x)$ and $g(x)$ have the same domain: $[0, \infty)$.

Would you like further details on any part? Here are some additional questions to deepen understanding:

1. Why does the function $f(x)$ in part (a) exclude $x = -1$?
2. What general rules apply for determining the domain of a rational function?
3. How does the square root affect the domain of functions involving $\sqrt{x}$?
4. Can a function have a domain in complex numbers, and how would that differ from real domains?
5. How would the domains change if $f(x)$ and $g(x)$ included additional transformations?

Tip: When dealing with rational functions, always check for values that make the denominator zero as they often restrict the domain.