Pick three of the operations that can be performed on functions and explain the algebra and geometry behind each operation.

Here are three common operations that can be performed on functions: **addition**, **composition**, and **differentiation**. Each operation affects both the algebraic expression of the function and its geometric graph.

### 1. **Addition of Functions**
**Algebra**: 
If we have two functions, \( f(x) \) and \( g(x) \), their addition is defined as:
\[
(f + g)(x) = f(x) + g(x)
\]
This operation simply adds the outputs of \( f \) and \( g \) for each input \( x \).

**Geometry**:
Graphically, the addition of two functions results in a new curve where the \( y \)-coordinate at each point is the sum of the \( y \)-coordinates of \( f \) and \( g \) at the same \( x \)-value. For instance, if \( f(x) \) and \( g(x) \) are both increasing, their sum will also be increasing but will stretch vertically. If one function is increasing while the other is decreasing, the sum may level out or cross the \( x \)-axis, depending on their relative magnitudes.

### 2. **Composition of Functions**
**Algebra**:
The composition of two functions \( f \) and \( g \), written as \( f(g(x)) \), means applying \( g(x) \) first and then applying \( f \) to the result:
\[
(f \circ g)(x) = f(g(x))
\]
In this operation, the input \( x \) is first transformed by \( g(x) \), and the result of that is then transformed by \( f \).

**Geometry**:
Geometrically, composition transforms the graph of one function by applying the second function. For example, if \( g(x) \) stretches or compresses the input \( x \)-values, then \( f(g(x)) \) will reflect those changes in the output values. The shape of the final graph can be very different from either original graph, depending on how \( g(x) \) modifies the input before passing it to \( f \).

### 3. **Differentiation of Functions**
**Algebra**:
The derivative of a function \( f(x) \) is denoted \( f'(x) \) and is defined as the limit of the average rate of change of \( f \) as \( x \) approaches a particular value. Mathematically:
\[
f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}
\]
This measures how \( f \) changes as \( x \) changes—specifically, the slope of the function at each point.

**Geometry**:
Geometrically, the derivative represents the slope of the tangent line to the graph of \( f(x) \) at any given point. If \( f'(x) > 0 \), the graph is increasing at \( x \); if \( f'(x) < 0 \), the graph is decreasing. A derivative of zero indicates a flat or stationary point, often corresponding to a local maximum, minimum, or a point of inflection.

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Would you like further details on these operations, or do you have any questions?

Here are 5 related questions you might consider:
1. How does the subtraction of functions differ from addition in both algebraic and geometric terms?
2. What happens when you compose a function with itself, i.e., \( f(f(x)) \)?
3. How does the chain rule relate to the composition of functions in differentiation?
4. What is the geometric interpretation of higher-order derivatives?
5. How do horizontal and vertical shifts of functions relate to function addition and transformation?

**Tip**: Always check the domain of the functions before performing operations, as operations like composition may restrict the domain of the resulting function.

Explore the operations of addition, composition, and differentiation on functions. Understand their algebraic formulas and geometric interpretations with detailed examples. Ideal for high school students in Grades 10-12.

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