To find the places where the graph of the function $y = 3^x - x^3$ crosses the X-axis within the range $[0, 4]$ in steps of 0.2, we can proceed with the following steps:

1. Evaluate the function at each point in the range:

We calculate the value of $y = 3^x - x^3$ for $x$ in the range $[0, 4]$ with steps of 0.2.

2. Identify where $y = 0$:

The graph crosses the X-axis where $y = 0$. We need to find the corresponding $x$ values where this occurs.

Let's calculate:

\hline x & y = 3^x - x^3 \\ \hline 0.0 & 3^0 - 0^3 = 1 - 0 = 1 \\ 0.2 & 3^{0.2} - 0.2^3 \approx 1.2457 - 0.008 \approx 1.2377 \\ 0.4 & 3^{0.4} - 0.4^3 \approx 1.5518 - 0.064 \approx 1.4878 \\ 0.6 & 3^{0.6} - 0.6^3 \approx 1.9332 - 0.216 \approx 1.7172 \\ 0.8 & 3^{0.8} - 0.8^3 \approx 2.4082 - 0.512 \approx 1.8962 \\ 1.0 & 3^{1.0} - 1.0^3 = 3 - 1 = 2 \\ 1.2 & 3^{1.2} - 1.2^3 \approx 3.5742 - 1.728 \approx 1.8462 \\ 1.4 & 3^{1.4} - 1.4^3 \approx 4.2768 - 2.744 \approx 1.5328 \\ 1.6 & 3^{1.6} - 1.6^3 \approx 5.1397 - 4.096 \approx 1.0437 \\ 1.8 & 3^{1.8} - 1.8^3 \approx 6.2031 - 5.832 \approx 0.3711 \\ 2.0 & 3^{2.0} - 2.0^3 = 9 - 8 = 1 \\ 2.2 & 3^{2.2} - 2.2^3 \approx 10.384 - 10.648 \approx -0.264 \\ 2.4 & 3^{2.4} - 2.4^3 \approx 12.245 - 13.824 \approx -1.579 \\ 2.6 & 3^{2.6} - 2.6^3 \approx 14.609 - 17.576 \approx -2.967 \\ 2.8 & 3^{2.8} - 2.8^3 \approx 17.515 - 21.952 \approx -4.437 \\ 3.0 & 3^{3.0} - 3.0^3 = 27 - 27 = 0 \\ 3.2 & 3^{3.2} - 3.2^3 \approx 33.696 - 32.768 \approx 0.928 \\ 3.4 & 3^{3.4} - 3.4^3 \approx 41.905 - 39.304 \approx 2.601 \\ 3.6 & 3^{3.6} - 3.6^3 \approx 52.212 - 46.656 \approx 5.556 \\ 3.8 & 3^{3.8} - 3.8^3 \approx 65.794 - 54.872 \approx 10.922 \\ 4.0 & 3^{4.0} - 4.0^3 = 81 - 64 = 17 \\ \hline \end{array}$$ ### 3. **Determine where the graph crosses the X-axis:** The graph crosses the X-axis where $$y = 0$$. From the table: - Between $$x = 2.0$$ and $$x = 2.2$$, $$y$$ changes from positive to negative. - At $$x = 3.0$$, $$y = 0$$. ### Conclusion: The graph of the function $$y = 3^x - x^3$$ crosses the X-axis approximately between $$x = 2.0$$ and $$x = 2.2$$, and exactly at $$x = 3.0$$. --- Would you like further details or have any questions? Here are some related questions for further exploration: 1. How can we more precisely find the exact value where the function $$y = 3^x - x^3$$ crosses the X-axis between 2.0 and 2.2? 2. What happens to the function $$y = 3^x - x^3$$ as $$x$$ increases beyond 4? 3. How does the behavior of the function change when the base of the exponential function is different, say $$y = 2^x - x^3$$? 4. What are the roots of the equation $$3^x = x^3$$? 5. How would you graph this function using a graphing calculator or software? **Tip:** When analyzing functions like $$y = 3^x - x^3$$, checking values at small intervals helps to capture the behavior more accurately, especially near critical points like where the function crosses the X-axis.