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The given function is \( g(x) = 2x^2 + 3 \). The question asks whether the inverse of this function, \( g^{-1}(x) \), exists.

### To check if the inverse exists:

For a function to have an inverse, it must be **bijective**:
1. **Injective** (One-to-one): Every input must map to a unique output.
2. **Surjective** (Onto): The function must cover the entire range of possible outputs.

Since \( g(x) = 2x^2 + 3 \) is a quadratic function, its graph is a parabola, and parabolas are not injective because they are symmetric. In other words, for any value of \( g(x) \), there are two possible values of \( x \) (one positive and one negative) that could map to the same output.

For example:
- If \( g(x_1) = g(x_2) \), then \( 2x_1^2 + 3 = 2x_2^2 + 3 \), which simplifies to \( x_1^2 = x_2^2 \). Therefore, \( x_1 = \pm x_2 \).

This means the function is not one-to-one (injective), and thus **it does not have an inverse** without further restrictions, like limiting the domain.

### Conclusion:
The inverse function \( g^{-1}(x) \) **does not exist** as the function \( g(x) = 2x^2 + 3 \) is not injective.

Answer: **False**

Would you like further details on this or any related concept? Here are 5 follow-up questions for deeper understanding:

1. What is the difference between injective and surjective functions?
2. How can you restrict the domain of a function to make it invertible?
3. Can you calculate the inverse of a quadratic function if the domain is restricted?
4. How do you graphically determine whether a function has an inverse?
5. What are the necessary conditions for a function to be considered bijective?

**Tip:** Always remember to check if a function is one-to-one before determining if an inverse exists.

Given a function g(x) = 2x^2 + 3, then an inverse of the function defined as g^{-1}(x) exists. True or False?

Find out whether the inverse of the quadratic function g(x) = 2x^2 + 3 exists by exploring injectivity and the conditions required for a function to have an inverse. Suitable for students in Grades 9-12.

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