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.