To graph the function \( g(x) = \sqrt{x^3 + 8} \), we begin by determining its domain. Since the expression is under a square root, it must be non-negative, leading to the inequality \( x^3 + 8 \geq 0 \). Solving this gives \( x \geq -2 \), indicating that the domain of the function is \( [-2, \infty) \).

Next, we find the x- and y-intercepts. For the x-intercept, we set the function equal to zero: \( \sqrt{x^3 + 8} = 0 \). This simplifies to \( x^3 + 8 = 0 \), resulting in \( x = -2 \). For the y-intercept, we evaluate \( g(0) = \sqrt{0^3 + 8} = \sqrt{8} = 2\sqrt{2} \approx 2.83 \). Thus, the intercepts are at \( (-2, 0) \) and \( (0, 2\sqrt{2}) \).

As we analyze the function for asymptotes, we find that there are none, as square root functions do not exhibit asymptotic behavior. To check for symmetry, we substitute \( -x \) into the function: \( g(-x) = \sqrt{(-x)^3 + 8} = \sqrt{-x^3 + 8} \). This does not equal \( g(x) \) or \( -g(x) \), indicating that the function is neither even nor odd.

Next, we explore the behavior of the function by finding its first derivative. Rewriting the function as \( g(x) = (x^3 + 8)^{1/2} \), we apply the power rule and chain rule to find:

\( g'(x) = \frac{1}{2}(x^3 + 8)^{-1/2} \cdot 3x^2 = \frac{3x^2}{2\sqrt{x^3 + 8}}. \)

To find critical points, we set the numerator equal to zero: \( 3x^2 = 0 \), yielding \( x = 0 \). The derivative does not exist at \( x = -2 \), but since this point is not in the domain, we only consider \( x = 0 \). The intervals to test for increasing or decreasing behavior are \( [-2, 0) \) and \( (0, \infty) \). Testing values in these intervals shows that \( g'(x) > 0 \) for both intervals, indicating that the function is increasing throughout its domain.

To analyze concavity, we compute the second derivative using the quotient rule. The second derivative simplifies to:

\( g''(x) = \frac{3x(x^3 + 32)}{4(x^3 + 8)\sqrt{x^3 + 8}}. \)

Setting the numerator equal to zero gives potential inflection points at \( x = 0 \) and \( x^3 + 32 = 0 \) (the latter being outside the domain). The second derivative does not exist at \( x = -2 \). Testing intervals around these points reveals that the function is concave down on \( [-2, 0) \) and concave up on \( (0, \infty) \).

In summary, the function is increasing throughout its domain, with a point of inflection at \( x = 0 \). The graph starts at \( (-2, 0) \), increases while concave down until \( x = 0 \), and then continues to increase while concave up. The final sketch reflects these characteristics, showing a smooth curve that transitions from concave down to concave up at the inflection point.

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