Here’s a table of values for both \( g(x) = \sqrt{x} \) and its inverse \( g^{-1}(x) = x^2 \) at \( x = -3, -2, -1, 0, 1, 2, 3 \).

However, note that \( g(x) = \sqrt{x} \) is only defined for non-negative values of \( x \). So for negative values, the square root function is undefined. For the inverse function \( g^{-1}(x) = x^2 \), there are no such restrictions.

| \( x \)   | \( g(x) = \sqrt{x} \) | \( g^{-1}(x) = x^2 \) |
|:--------:|:--------------------:|:---------------------:|
| -3       | Undefined            | 9                     |
| -2       | Undefined            | 4                     |
| -1       | Undefined            | 1                     |
|  0       | 0                    | 0                     |
|  1       | 1                    | 1                     |
|  2       | 1.414                | 4                     |
|  3       | 1.732                | 9                     |

Values are rounded for \( g(x) = \sqrt{x} \), as the square roots are often irrational. For negative values of \( x \), the square root function is undefined, while the inverse function \( g^{-1}(x) = x^2 \) is valid for all real numbers. Graph that table 

Here is the graph of \( g(x) = \sqrt{x} \) and its inverse \( g^{-1}(x) = x^2 \) based on the table of values. The blue curve represents \( g(x) = \sqrt{x} \), which is only defined for non-negative values of \( x \), while the red curve represents \( g^{-1}(x) = x^2 \), which is defined for all real numbers.

Would you like further explanation or any specific details?

Here are some related questions to explore:
1. What happens when you reflect the graph of a function over the line \( y = x \)?
2. How can we generalize the concept of inverses for other functions?
3. Why is \( g(x) = \sqrt{x} \) undefined for negative values of \( x \)?
4. Can the square root function have complex values for negative inputs?
5. How do we determine if a function has an inverse?

**Tip:** The inverse of a function essentially "reverses" the operation of the original function, which is why their graphs are symmetric across the line \( y = x \).

Here’s a table of values for both g(x) = √x and its inverse g⁻¹(x) = x² at x = -3, -2, -1, 0, 1, 2, 3. However, note that g(x) = √x is only defined for non-negative values of x. So for negative values, the square root function is undefined. For the inverse function g⁻¹(x) = x², there are no such restrictions.

Explore the properties of the function g(x) = √x and its inverse g⁻¹(x) = x². This problem covers key concepts in functions and inverse functions, including a detailed table of values for x = -3, -2, -1, 0, 1, 2, 3. Learn why g(x) is undefined for negative values of x, while the inverse is valid for all real numbers. Suitable for students in Grades 9-12.

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