To determine whether the given function [tex]\( y = e^{x^{8 + 2x}} \)[/tex] is an exponential function, let's carefully examine the structure of the function:
1. Understanding Exponential Functions:
- An exponential function typically has the form [tex]\( y = a \cdot e^{bx} \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( x \)[/tex] is the variable. In this form, the variable [tex]\( x \)[/tex] is in the exponent, and it is raised to a power linearly (or simply multiplied by a constant).
2. Analyzing the Given Function:
- The given function is [tex]\( y = e^{x^{8 + 2x}} \)[/tex].
- Here, the exponent is [tex]\( x^{8 + 2x} \)[/tex]. Unlike a standard exponential function where the exponent is linear in [tex]\( x \)[/tex], in this case, the exponent is not linear. It is instead a more complex function of [tex]\( x \)[/tex], as it involves [tex]\( x \)[/tex] raised to another function involving [tex]\( x \)[/tex].
3. Conclusion:
- Since the exponent [tex]\( x^{8 + 2x} \)[/tex] is not a linear function of [tex]\( x \)[/tex], the given function [tex]\( y = e^{x^{8 + 2x}} \)[/tex] does not fit the standard form of an exponential function.
Therefore, the given function is not an exponential function.
