➕ Definition of Addition of Functions
The addition of two functions, denoted as $(f + g)(x)$, is defined as the sum of the individual functions $f(x)$ and $g(x)$. Mathematically, it's expressed as:
$(f + g)(x) = f(x) + g(x)$
- ➕Key Principle: To add functions, simply add their corresponding expressions.
- 📝Example: If $f(x) = x^2$ and $g(x) = 2x + 1$, then $(f + g)(x) = x^2 + 2x + 1$.
- 💡Real-world application: Combining cost functions in business to find the total cost.
➖ Definition of Subtraction of Functions
The subtraction of two functions, denoted as $(f - g)(x)$, is defined as the difference between the functions $f(x)$ and $g(x)$. The order matters! It's expressed as:
$(f - g)(x) = f(x) - g(x)$
- ➖Key Principle: Subtract the second function's expression from the first. Pay attention to signs!
- 📝Example: If $f(x) = 3x$ and $g(x) = x - 2$, then $(f - g)(x) = 3x - (x - 2) = 2x + 2$.
- 📈Real-world application: Calculating profit by subtracting the cost function from the revenue function.
✖️ Definition of Multiplication of Functions
The multiplication of two functions, denoted as $(f \cdot g)(x)$ or $(fg)(x)$, is defined as the product of the individual functions $f(x)$ and $g(x)$. It's expressed as:
$(f \cdot g)(x) = f(x) \cdot g(x)$
- ✖️Key Principle: Multiply the expressions of the two functions.
- 📝Example: If $f(x) = x + 1$ and $g(x) = x - 1$, then $(f \cdot g)(x) = (x + 1)(x - 1) = x^2 - 1$.
- 📐Real-world application: Finding the area of a rectangle where the length and width are defined as functions.
➗ Definition of Division of Functions
The division of two functions, denoted as $(f / g)(x)$, is defined as the quotient of the functions $f(x)$ and $g(x)$, where $g(x)$ cannot be zero. It's expressed as:
$(f / g)(x) = \frac{f(x)}{g(x)}$, where $g(x) \neq 0$
- ➗Key Principle: Divide the expression of the first function by the expression of the second function. Always consider the domain!
- 📝Example: If $f(x) = x^2 - 1$ and $g(x) = x + 1$, then $(f / g)(x) = \frac{x^2 - 1}{x + 1} = x - 1$, where $x \neq -1$.
- 🌍Real-world application: Calculating average cost by dividing the total cost function by the number of units.