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The given problem asks to prove that \( f(a) + f(b) = f(a+b) \), given the following definition of the function \( f(x) \):

\[
f(x) = b \frac{x - a}{b - a} + a \frac{x - b}{a - b}
\]

### Step 1: Compute \( f(a) \)
Substitute \( x = a \) into the given function:

\[
f(a) = b \frac{a - a}{b - a} + a \frac{a - b}{a - b}
\]

Simplifying the terms:

\[
f(a) = b \cdot 0 + a \cdot 1 = a
\]

So, \( f(a) = a \).

### Step 2: Compute \( f(b) \)
Now, substitute \( x = b \) into the function:

\[
f(b) = b \frac{b - a}{b - a} + a \frac{b - b}{a - b}
\]

Simplifying the terms:

\[
f(b) = b \cdot 1 + a \cdot 0 = b
\]

Thus, \( f(b) = b \).

### Step 3: Compute \( f(a+b) \)
Now, substitute \( x = a + b \) into the function:

\[
f(a+b) = b \frac{a + b - a}{b - a} + a \frac{a + b - b}{a - b}
\]

Simplifying the terms:

\[
f(a+b) = b \frac{b}{b - a} + a \frac{a}{a - b}
\]

This becomes:

\[
f(a+b) = \frac{b^2}{b - a} + \frac{a^2}{a - b}
\]

Note that \( \frac{a^2}{a - b} = -\frac{a^2}{b - a} \), so we can rewrite this as:

\[
f(a+b) = \frac{b^2 - a^2}{b - a}
\]

Using the identity \( b^2 - a^2 = (b - a)(b + a) \), we get:

\[
f(a+b) = b + a
\]

Thus, \( f(a+b) = a + b \).

### Step 4: Conclusion
We have shown that:

\[
f(a) + f(b) = a + b
\]

And:

\[
f(a+b) = a + b
\]

Therefore, \( f(a) + f(b) = f(a+b) \).

Would you like further explanation or more details on any of the steps?

Here are 5 follow-up questions to deepen understanding:
1. What mathematical identity was used to simplify \( b^2 - a^2 \)?
2. How does the form of \( f(x) \) relate to the concept of interpolation?
3. Could the result be generalized to functions other than linear ones?
4. What happens if \( a = b \) in the given function \( f(x) \)?
5. How would the proof change if we considered complex numbers for \( a \) and \( b \)?

**Tip**: Always check if expressions can be factored to simplify complicated terms during algebraic manipulation.

If f(x) = b * (x - a) / (b - a) + a * (x - b) / (a - b), then show that f(a) + f(b) = f(a + b).

Learn how to prove that f(a) + f(b) = f(a + b) using algebraic manipulation. The problem explores functional equations, interpolation, and the properties of linear functions. Ideal for advanced high school students in Grades 11-12.

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