๐ Arithmetic Operations on Functions: The Basics
Arithmetic operations on functions involve combining functions using basic mathematical operations like addition, subtraction, multiplication, and division. Essentially, you're performing these operations on the outputs of the functions for a given input.
- โ Addition: $(f+g)(x) = f(x) + g(x)$
- โ Subtraction: $(f-g)(x) = f(x) - g(x)$
- โ๏ธ Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$
- โ Division: $(f/g)(x) = f(x) / g(x)$, where $g(x) \neq 0$
๐งฎ Composition of Functions: The Basics
Composition of functions, on the other hand, involves plugging one function into another. The output of one function becomes the input of the other. It's like a mathematical assembly line!
- ๐ฏ Definition: $(f \circ g)(x) = f(g(x))$. This means you first evaluate $g(x)$, and then you plug that result into the function $f$.
๐ Arithmetic Operations vs. Composition: A Detailed Comparison
| Feature |
Arithmetic Operations |
Composition of Functions |
| What it is |
Combining function outputs using $+$, $-$, $\times$, or $/$. |
Using the output of one function as the input of another. |
| Notation |
$(f+g)(x)$, $(f-g)(x)$, $(f \cdot g)(x)$, $(f/g)(x)$ |
$(f \circ g)(x)$ or $f(g(x))$ |
| Order Matters? |
Addition and multiplication are commutative (order doesn't matter), but subtraction and division are not. |
Yes, order *always* matters. $f(g(x))$ is generally not the same as $g(f(x))$. |
| Example |
If $f(x) = x^2$ and $g(x) = x + 1$, then $(f+g)(x) = x^2 + x + 1$. |
If $f(x) = x^2$ and $g(x) = x + 1$, then $(f \circ g)(x) = (x+1)^2 = x^2 + 2x + 1$. |
๐ Key Takeaways
- ๐ก Arithmetic: Think of these as direct combinations using basic math.
- โ๏ธ Composition: Think of this as a chain reaction where one function feeds into another.
- โ๏ธ Order: Pay close attention to the order, especially with composition!