2. \( f(x)=x^{2} \sqrt{x+1} \)

The function \( f(x) = x^2 \sqrt{x + 1} \) showcases how polynomials combine with radical functions to create intriguing behaviors. As \( x \) becomes large, the function predominantly resembles \( x^{2.5} \) due to the square root, which causes rapid growth. However, the behavior changes dramatically as \( x \) approaches negative values because the square root becomes imaginary for \( x < -1 \), emphasizing the importance of the domain in function analysis. When evaluating or graphing this function, remember to consider its domain carefully. The radical requires \( x + 1 \geq 0\), meaning \( x \geq -1 \). A common mistake is to overlook this restriction, potentially leading to confusion about where the function is defined. Always check for values that might make a radical or denominator undefined—it's crucial for accurate results!