Functions in Algebra

Study Notes

Types of Functions

Linear Functions: A function in which the highest power of the variable (usually x) is 1. Examples: f(x) = 2x + 3, f(x) = x - 4.
Quadratic Functions: A function in which the highest power of the variable (usually x) is 2. Examples: f(x) = x^2 + 4x + 4, f(x) = x^2 - 3x - 2.
Polynomial Functions: A function consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Examples: f(x) = x^4 + 2x^3 - 3x^2 + x - 1, f(x) = x^2 + 2x - 3.
Rational Functions: A function that can be expressed as the ratio of two polynomial functions. Examples: f(x) = (x + 1) / (x - 1), f(x) = (x^2 + 2) / (x - 2).
Exponential Functions: A function in which the variable is in the exponent. Examples: f(x) = 2^x, f(x) = 3^(2x).
Logarithmic Functions: A function that is the inverse of an exponential function. Examples: f(x) = log2(x), f(x) = ln(x).

Function Operations

Function Addition: (f + g)(x) = f(x) + g(x)
Function Subtraction: (f - g)(x) = f(x) - g(x)
Function Multiplication: (f × g)(x) = f(x) × g(x)
Function Division: (f ÷ g)(x) = f(x) ÷ g(x), where g(x) ≠ 0

Function Properties

Domain: The set of input values for which the function is defined.
Range: The set of output values of the function.
Even Function: A function that satisfies f(-x) = f(x) for all x in its domain.
Odd Function: A function that satisfies f(-x) = -f(x) for all x in its domain.

Graphing Functions

X-Intercept: The point at which the graph intersects the x-axis.
Y-Intercept: The point at which the graph intersects the y-axis.
Asymptotes: Lines that the graph approaches as the input values increase or decrease without bound.
Maxima and Minima: The highest and lowest points of the graph, respectively.

Description

Test your understanding of different types of functions, including linear, quadratic, polynomial, rational, exponential, and logarithmic functions. Learn about function operations, properties, and graphing functions.