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The problem asks to sketch the graph of the function \( g(x) = x^3 - x \).

To analyze and sketch this function, we can follow these steps:

1. **Identify Intercepts:**
   - **x-intercepts:** Solve \( g(x) = 0 \) by setting \( x^3 - x = 0 \).
     \[
     x(x^2 - 1) = 0
     \]
     This factors further as:
     \[
     x(x - 1)(x + 1) = 0
     \]
     Therefore, the x-intercepts are \( x = 0 \), \( x = 1 \), and \( x = -1 \).

   - **y-intercept:** Substitute \( x = 0 \) into \( g(x) \) to find \( g(0) = 0 \), so the y-intercept is also at \( (0, 0) \).

2. **Find the Derivative for Critical Points:**
   - To find critical points where the slope is zero or undefined, we differentiate \( g(x) \):
     \[
     g'(x) = 3x^2 - 1
     \]
   - Set \( g'(x) = 0 \) to find critical points:
     \[
     3x^2 - 1 = 0
     \]
     \[
     x^2 = \frac{1}{3}
     \]
     \[
     x = \pm \frac{1}{\sqrt{3}}
     \]
     These values give two critical points at \( x = \frac{1}{\sqrt{3}} \) and \( x = -\frac{1}{\sqrt{3}} \).

3. **Determine Concavity and Inflection Points:**
   - Compute the second derivative \( g''(x) \) to determine concavity:
     \[
     g''(x) = 6x
     \]
   - Set \( g''(x) = 0 \) to find possible inflection points:
     \[
     6x = 0 \Rightarrow x = 0
     \]
     So, \( x = 0 \) is an inflection point.

4. **Behavior at Critical Points and Inflection Point:**
   - At \( x = \pm \frac{1}{\sqrt{3}} \), check the sign of \( g'(x) \) around these points to determine if they are local minima or maxima.
   - Since \( g''(x) > 0 \) for \( x > 0 \) and \( g''(x) < 0 \) for \( x < 0 \), the concavity changes at \( x = 0 \), confirming it as an inflection point.

5. **Sketching the Graph:**
   - Plot the intercepts at \( (-1, 0) \), \( (0, 0) \), and \( (1, 0) \).
   - Show the local maximum and minimum near \( x = \pm \frac{1}{\sqrt{3}} \).
   - Illustrate the inflection point at \( x = 0 \), where the curve changes concavity.

Would you like me to create a visual plot of this function? Or do you have questions about any specific step? 

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Here are five related questions for further exploration:
1. How would the graph change if the function were \( g(x) = x^3 + x \) instead?
2. What is the significance of inflection points in graphing polynomial functions?
3. How do local extrema relate to the derivative of a function?
4. How does concavity influence the shape of the graph?
5. Could you determine behavior at infinity for functions of this degree?

**Tip:** To better understand the behavior near critical points, examine both the first and second derivatives around those points.

Explore the graphing of the polynomial function g(x) = x^3 - x. This problem covers calculus concepts, including finding intercepts, critical points, inflection points, and analyzing concavity. Suitable for high school calculus students in Grades 11-12.

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