Mathematical functions are formal relationships between sets, where each element in the first set (called the domain) maps to exactly one element in the second set (called the codomain). Functions express how one quantity is determined by another.
Types of Mathematical Functions
- Algebraic functions: Defined as solutions to polynomial equations including polynomials, rational functions (ratios of polynomials), and root functions like square or cube roots.
- Elementary transcendental functions: Include exponential functions, logarithms, and trigonometric functions like sine and cosine, which cannot be represented as algebraic functions.
- Polynomial functions: Functions composed of sums of powers of the variable with constant coefficients, including special cases like linear, quadratic, cubic, quartic, and quintic functions.
- Real functions: Functions with real number inputs and outputs, often continuous and differentiable forms important in calculus.
- Multivariate functions: Functions dependent on multiple variables, common in real-world modeling where outputs depend on several inputs.
Function Properties and Uses
- Functions can be represented algebraically, graphically, or verbally.
- The function's graph passes the vertical line test to confirm validity.
- Operations on functions include sums, differences, products, and quotients, with domains adjusted accordingly.
- Functions model real-life phenomena, such as area calculations and physical relationships.
Functions are fundamental in mathematics for understanding and describing variable relationships across many fields.
