If f(x) = eâx+b, Prove that eb. f(x+y) = f(x) . f(y)

To prove the identity \( e^b \cdot f(x + y) = f(x) \cdot f(y) \), where \( f(x) = e^{ax + b} \), we will proceed with the following steps.

### Step 1: Define the given function
We are given the function:

\[
f(x) = e^{ax + b}
\]

where \( a \) and \( b \) are constants.

### Step 2: Express \( f(x + y) \)
Now, let's express \( f(x + y) \):

\[
f(x + y) = e^{a(x + y) + b} = e^{ax + ay + b}
\]

### Step 3: Multiply \( e^b \) by \( f(x + y) \)
Next, we multiply both sides of the equation \( f(x + y) \) by \( e^b \):

\[
e^b \cdot f(x + y) = e^b \cdot e^{ax + ay + b}
\]

Using the property of exponents \( e^m \cdot e^n = e^{m + n} \), we get:

\[
e^b \cdot f(x + y) = e^{b + ax + ay + b} = e^{2b + ax + ay}
\]

### Step 4: Express the product \( f(x) \cdot f(y) \)
Now, let's compute the product \( f(x) \cdot f(y) \):

\[
f(x) \cdot f(y) = e^{ax + b} \cdot e^{ay + b}
\]

Again, using the exponent multiplication rule, we get:

\[
f(x) \cdot f(y) = e^{ax + b + ay + b} = e^{ax + ay + 2b}
\]

### Step 5: Compare both sides
We have:

\[
e^b \cdot f(x + y) = e^{2b + ax + ay}
\]

and

\[
f(x) \cdot f(y) = e^{ax + ay + 2b}
\]

Clearly, both sides are equal. Thus, we have shown that:

\[
e^b \cdot f(x + y) = f(x) \cdot f(y)
\]

### Conclusion:
We have proven that \( e^b \cdot f(x + y) = f(x) \cdot f(y) \) for the function \( f(x) = e^{ax + b} \).

---

Would you like any further clarification on this proof or have any other questions? 

Here are 5 related questions for deeper exploration:

1. How would the proof change if \( f(x) = e^{ax + c} \), where \( c \) is a constant different from \( b \)?
2. What happens if the base of the exponential function is not \( e \) but some other constant \( k \)?
3. How would you derive a similar identity if the function were \( f(x) = e^{a(x + b)} \)?
4. What is the general form of functions that satisfy \( f(x + y) = f(x) \cdot f(y) \)?
5. How can this identity be used in solving differential equations involving exponential functions?

**Tip:** When dealing with exponential functions, always remember the rule \( e^m \cdot e^n = e^{m + n} \), which is key for simplifying expressions involving exponentials.

If f(x) = e^{ax+b}, prove that e^b . f(x+y) = f(x) . f(y)

Learn how to prove the exponential identity e^b * f(x + y) = f(x) * f(y) for the function f(x) = e^{ax + b}. This problem explores key concepts such as the properties of exponential functions and exponentiation rules, suitable for high school students in Grades 11-12.

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