Algebra of functions
Algebra of Functions
Algebra of functions refers to the operations that can be performed on functions, including addition, subtraction, multiplication, division, and composition. These operations allow us to create new functions from existing ones. Understanding the algebra of functions is crucial for solving complex problems in calculus, differential equations, and other areas of mathematics.
Operations on Functions
Given two functions, $f(x)$ and $g(x)$, we can define the following operations:
Addition
The sum of $f(x)$ and $g(x)$ is a new function $h(x)$ defined by:
$$ h(x) = (f + g)(x) = f(x) + g(x) $$
Subtraction
The difference of $f(x)$ and $g(x)$ is a new function $h(x)$ defined by:
$$ h(x) = (f - g)(x) = f(x) - g(x) $$
Multiplication
The product of $f(x)$ and $g(x)$ is a new function $h(x)$ defined by:
$$ h(x) = (f \cdot g)(x) = f(x) \cdot g(x) $$
Division
The quotient of $f(x)$ and $g(x)$, where $g(x) \neq 0$, is a new function $h(x)$ defined by:
$$ h(x) = \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} $$
Composition
The composition of $f(x)$ and $g(x)$, denoted by $f \circ g$, is a new function $h(x)$ defined by:
$$ h(x) = (f \circ g)(x) = f(g(x)) $$
Important Points and Differences
| Operation | Symbol | Definition | Domain | Example |
|---|---|---|---|---|
| Addition | $f + g$ | $(f + g)(x) = f(x) + g(x)$ | $x$ such that $x$ is in both domains of $f$ and $g$ | If $f(x) = x^2$ and $g(x) = 3x$, then $(f + g)(x) = x^2 + 3x$ |
| Subtraction | $f - g$ | $(f - g)(x) = f(x) - g(x)$ | $x$ such that $x$ is in both domains of $f$ and $g$ | If $f(x) = x^2$ and $g(x) = 3x$, then $(f - g)(x) = x^2 - 3x$ |
| Multiplication | $f \cdot g$ | $(f \cdot g)(x) = f(x) \cdot g(x)$ | $x$ such that $x$ is in both domains of $f$ and $g$ | If $f(x) = x^2$ and $g(x) = 3x$, then $(f \cdot g)(x) = x^2 \cdot 3x$ |
| Division | $\frac{f}{g}$ | $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$ | $x$ such that $x$ is in both domains of $f$ and $g$, and $g(x) \neq 0$ | If $f(x) = x^2$ and $g(x) = 3x$, then $\left(\frac{f}{g}\right)(x) = \frac{x^2}{3x}$, $x \neq 0$ |
| Composition | $f \circ g$ | $(f \circ g)(x) = f(g(x))$ | $x$ such that $g(x)$ is in the domain of $f$ | If $f(x) = x^2$ and $g(x) = 3x$, then $(f \circ g)(x) = (3x)^2$ |
Examples
Let's consider two functions $f(x) = 2x + 1$ and $g(x) = x^2 - 4$ to illustrate the algebra of functions.
Example 1: Addition
$$ (f + g)(x) = f(x) + g(x) = (2x + 1) + (x^2 - 4) = x^2 + 2x - 3 $$
Example 2: Subtraction
$$ (f - g)(x) = f(x) - g(x) = (2x + 1) - (x^2 - 4) = -x^2 + 2x + 5 $$
Example 3: Multiplication
$$ (f \cdot g)(x) = f(x) \cdot g(x) = (2x + 1)(x^2 - 4) = 2x^3 - 8x + x^2 - 4 $$
Example 4: Division
$$ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{2x + 1}{x^2 - 4} \quad \text{(for $x \neq \pm 2$)} $$
Example 5: Composition
$$ (f \circ g)(x) = f(g(x)) = f(x^2 - 4) = 2(x^2 - 4) + 1 = 2x^2 - 8 + 1 = 2x^2 - 7 $$
Conclusion
The algebra of functions allows us to combine functions in various ways to create new functions. Each operation has its own set of rules and domain considerations. It's important to remember that the domain of the resulting function must be restricted to values that are valid for both original functions, especially in the case of division and composition. Understanding these operations is fundamental for further studies in mathematics, particularly in calculus where these concepts are applied to analyze and solve real-world problems.