The function given in the image is $f(x) = e^{2x+1}$, and it asks to determine whether this function exists.

Analyzing the Function:

The function is an exponential function of the form $f(x) = e^{g(x)}$, where $g(x) = 2x + 1$.

Existence of the Function:

• Exponential functions of the form $e^{g(x)}$ exist for all real values of $g(x)$, and consequently for all real values of $x$.
• Since $g(x) = 2x + 1$ is a linear function, it is defined for all $x \in \mathbb{R}$.

Conclusion:

• The function $f(x) = e^{2x+1}$ exists for all real values of $x$.

Would you like more details, or do you have any other questions?

Here are some related questions to consider:

1. How do we differentiate $f(x) = e^{2x+1}$?
2. What is the integral of $f(x) = e^{2x+1}$?
3. How does the graph of $f(x) = e^{2x+1}$ compare to that of $f(x) = e^x$?
4. What is the behavior of $f(x) = e^{2x+1}$ as $x$ approaches infinity?
5. How would you solve $f(x) = 10$ for $x$?

Tip: Exponential functions like $e^{x}$ are always positive and grow rapidly as $x$ increases.