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The question in your image asks to find the point of intersection of the graphs of the functions \( g(x) = x^2 + 2 \) and \( h(x) = 3.5 - x^3 \).

To find the point of intersection of two functions, we need to set the functions equal to each other and solve for \( x \):

\[
g(x) = h(x)
\]

\[
x^2 + 2 = 3.5 - x^3
\]

This gives us the equation:

\[
x^3 + x^2 - 3.5 + 2 = 0
\]

\[
x^3 + x^2 - 1.5 = 0
\]

Now, we can solve this cubic equation for \( x \). Let me compute that for you.The real solution to the equation \( x^3 + x^2 - 1.5 = 0 \) is approximately \( x = 0.891 \). The other two solutions are complex numbers, so they do not represent points of intersection on the real-number plane.

Now, let's find the corresponding \( y \)-value by substituting \( x = 0.891 \) into either function \( g(x) \) or \( h(x) \). I'll compute that next.The corresponding \( y \)-value at \( x = 0.891 \) is approximately \( y = 2.793 \).

Therefore, the point of intersection of the graphs of the functions \( g(x) = x^2 + 2 \) and \( h(x) = 3.5 - x^3 \) is approximately \( (0.891, 2.793) \).

Would you like more details on this solution or have any further questions?

Here are five related questions you might find useful:
1. How can you verify the point of intersection graphically?
2. What methods can be used to solve cubic equations besides factoring?
3. How can you determine if two functions have multiple points of intersection?
4. How do the graphs of cubic and quadratic functions typically differ in shape?
5. What are some real-world applications of finding points of intersection between functions?

**Tip:** When solving cubic equations, always check for both real and complex roots, as only real roots represent points of intersection in real-world contexts.

Find the point of intersection of the graphs of the functions g(x) = x^2 + 2 and h(x) = 3.5 - x^3.

Learn how to find the point of intersection between the functions g(x) = x^2 + 2 and h(x) = 3.5 - x^3. The solution involves setting the functions equal and solving a cubic equation for the x-coordinate, followed by substituting the x-value into one of the functions to find the corresponding y-coordinate. Suitable for high school students.

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