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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 \( a \).

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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.

Gegeben sei die Funktion f mit f(x) = b * a^x. Erläutere, wie sich der Funktionswert der Exponentialfunktion f gegenüber dem ursprünglichen Wert verändert, wenn man x um 1 vergrößert.

Explore how the value of the exponential function f(x) = b * a^x changes when the variable x is increased by 1. This step-by-step guide breaks down the behavior of exponential functions, explaining key concepts and formulas related to exponential growth. Suitable for high school students (Grades 10-12).

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