The given problem is in German and asks for an explanation of how the function value of the exponential function $f(x) = b \cdot a^x$ changes when $x$ is increased by 1. Let's break this down:

Function Overview:

The function is of the form: $f(x) = b \cdot a^x$ where $b$ and $a$ are constants, and $x$ is the variable.

Question Breakdown:

The question asks how the value of the function $f$ changes when $x$ is increased by 1. In mathematical terms, this means comparing $f(x+1)$ to $f(x)$.

Step-by-Step Solution:

1. Start with the function: $f(x) = b \cdot a^x$
2. Now, consider $f(x+1)$, which is the function value when $x$ is increased by 1: $f(x+1) = b \cdot a^{x+1} = b \cdot a^x \cdot a$
3. Compare $f(x+1)$ to $f(x)$: $f(x+1) = f(x) \cdot a$ This shows that when $x$ is increased by 1, the function value is multiplied by $a$.

Conclusion:

When $x$ increases by 1, the function value increases by a factor of $a$. This is characteristic of exponential growth, where each step in $x$ results in a multiplication by the base $a$.

Would you like a more detailed explanation on any part of this? Also, here are 5 related questions for further exploration:

1. What happens if $a = 1$ in the function $f(x) = b \cdot a^x$?
2. How does the function change if $a$ is less than 1 (i.e., a decay function)?
3. Can you explain the behavior of the function for negative values of $x$?
4. What is the significance of the constant $b$ in the function?
5. How would the derivative of $f(x) = b \cdot a^x$ look?

Tip: Understanding how exponential functions behave is crucial for modeling growth and decay processes, such as population growth or radioactive decay.