$$ \left(f+g\right)\left(x\right)=f\left(x\right) $$$$ +g\left(x\right) $$

The equation you've provided demonstrates the definition of the sum of two functions $f(x)$ and $g(x)$. Specifically, it states that the sum of the functions $f$ and $g$, evaluated at $x$, is equal to the sum of the individual function values at $x$.

Mathematically, this can be written as:

\[ (f+g)(x) = f(x) + g(x) \]

### Explanation

1. **$f(x)$**: This represents the value of the function $f$ at $x$.
2. **$g(x)$**: This represents the value of the function $g$ at $x$.
3. **$(f+g)(x)$**: This denotes the sum of the functions $f$ and $g$ at $x$.

When we add two functions $f$ and $g$, their sum $ (f + g) $ is a new function. The value of this new function at any point $x$ is simply the sum of the values of $f$ and $g$ at $x$.

### Example

If $ f(x) = 2x $ and $ g(x) = x^2 $, then:

\[ (f+g)(x) = f(x) + g(x) = 2x + x^2 \]

So, for any specific $x$, say $x = 3$:

\[ (f+g)(3) = f(3) + g(3) = 2(3) + (3)^2 = 6 + 9 = 15 \]

Would you like further details or have any questions about this? Here are some additional questions that expand on this topic:

1. What are some properties of function addition?
2. How do we subtract two functions?
3. What happens if we add a constant function to another function?
4. How do we multiply two functions?
5. What is the composition of two functions?
6. How does function addition apply to trigonometric functions?
7. Can you explain function addition in terms of a graphical representation?
8. How does adding functions affect their domains and ranges?

#### Tip: 
When working with functions, always check their domains to ensure that the operations you perform are valid for all inputs within those domains.

Explore the concept of function addition with clear explanations and examples. Learn how to add functions ( f ) and ( g ), and understand the mathematical definition (f+g)(x) = f(x) + g(x). Includes practical examples and implications of function addition.

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