To solve for [tex]\( F(x) \)[/tex], we need to carefully analyze and combine the components within the specified function. Let's break down the function into manageable parts.
Given:
[tex]\[ F(x) = \frac{e^{x^2 + 2x - 1}}{\sqrt{e^x - 1}} \][/tex]
### Step-by-Step Solution:
1. Understand the Numerator [tex]\( e^{x^2 + 2x - 1} \)[/tex]:
- The numerator consists of the exponential function [tex]\( e \)[/tex] raised to the power of a quadratic polynomial [tex]\( x^2 + 2x - 1 \)[/tex].
- [tex]\( e^{x^2 + 2x - 1} \)[/tex] represents an exponential growth where the exponent is a parabola opening upwards (since the coefficient of [tex]\( x^2 \)[/tex] is positive).
2. Understand the Denominator [tex]\( \sqrt{e^x - 1} \)[/tex]:
- The denominator is the square root of the expression [tex]\( e^x - 1 \)[/tex].
- Note that [tex]\( e^x \)[/tex] grows exponentially, and subtracting 1 makes sure the argument inside the square root is shifted by 1 unit downwards.
- For the square root to be real and defined, [tex]\( e^x - 1 \)[/tex] must be greater than 0, i.e., [tex]\( e^x > 1 \)[/tex], which is true for [tex]\( x > 0 \)[/tex].
3. Combining the Numerator and Denominator:
- The entire function [tex]\( F(x) \)[/tex] is the quotient of these two expressions.
- [tex]\( F(x) \)[/tex] combines the exponential growth in the numerator with a square root function in the denominator, which impacts the overall growth rate of the function.
### Recap of the Expression:
Putting it all together, our function is:
[tex]\[ F(x) = \frac{e^{x^2 + 2x - 1}}{\sqrt{e^x - 1}} \][/tex]
This is the simplified form of [tex]\( F(x) \)[/tex], showing the interplay between exponential and square root components.
### Conclusion:
The function we're given is complex due to its mixture of exponential and square root functions. However, understanding it step-by-step allows us to appreciate how it grows and behaves. For all practical purposes, [tex]\( e^{x^2 + 2x - 1} \)[/tex] in the numerator grows very quickly for increasing [tex]\( x \)[/tex], while the denominator [tex]\( \sqrt{e^x - 1} \)[/tex] also increases, albeit at a different rate.
So the final function after evaluating or simplifying as much as possible is:
[tex]\[ F(x) = \frac{e^{x^2 + 2x - 1}}{\sqrt{e^x - 1}} \][/tex]
