If f of x is equal to 10 x square then find f of X + y is equal to f of X + f of y

Given the function \( f(x) = 10x^2 \), you are asked to determine if the equation \( f(x + y) = f(x) + f(y) \) holds.

### Step 1: Compute \( f(x + y) \)
The function is given as \( f(x) = 10x^2 \). To find \( f(x + y) \), substitute \( x + y \) into the function:
\[
f(x + y) = 10(x + y)^2
\]

### Step 2: Expand \( (x + y)^2 \)
\[
(x + y)^2 = x^2 + 2xy + y^2
\]
Thus,
\[
f(x + y) = 10(x^2 + 2xy + y^2) = 10x^2 + 20xy + 10y^2
\]

### Step 3: Compute \( f(x) + f(y) \)
\[
f(x) + f(y) = 10x^2 + 10y^2
\]

### Step 4: Compare \( f(x + y) \) with \( f(x) + f(y) \)
\[
f(x + y) = 10x^2 + 20xy + 10y^2
\]
\[
f(x) + f(y) = 10x^2 + 10y^2
\]
Clearly, \( f(x + y) \) includes the term \( 20xy \), which is not present in \( f(x) + f(y) \).

### Conclusion:
Since \( f(x + y) \) is not equal to \( f(x) + f(y) \), the equation \( f(x + y) = f(x) + f(y) \) **does not hold** for the given function \( f(x) = 10x^2 \).

Would you like more details on this problem or have any questions? 

Here are five related questions:
1. What is a linear function, and why is it important for the property \( f(x + y) = f(x) + f(y) \)?
2. How does the function \( f(x) = 10x^2 \) differ from a linear function?
3. What type of function satisfies the equation \( f(x + y) = f(x) + f(y) \)?
4. Can a quadratic function ever satisfy \( f(x + y) = f(x) + f(y) \)?
5. What is the significance of the term \( 20xy \) in the expanded form of \( f(x + y) \)?

**Tip:** For a function to satisfy the condition \( f(x + y) = f(x) + f(y) \), it must be a linear function, meaning it can be written in the form \( f(x) = ax + b \).

Explore the properties of quadratic functions with a focus on f(x) = 10x^2. Determine if the equation f(x + y) = f(x) + f(y) holds true through detailed calculations and analysis. Suitable for advanced high school students studying quadratic functions.

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