Algebraic Functions

Basic Understanding of Algebraic Functions

• Algebraic functions are function that can be defined using algebraic operations.
• They involve variables, constants and operations – addition, subtraction, multiplication, division, and exponentiation.
• Examples include polynomial functions, rational functions, exponential functions, and so on.

Function Notation

• Function notation involves writing functions as f(x) or g(x), where f or g is the function, and x is the input.
• In an equation y = f(x), y is the output corresponding to an input x.

Evaluating Functions

• Function evaluation involves substitifying a specific value for the variable in the function.
• If you have a function f(x), by replacing x with a number, you can find the output or the value of the function.
• For example, if f(x) = 2x + 3, then f(2) = (22) + 3 = 7*.

Types of Functions

• Linear functions are functions of the form f(x) = mx + c where m and c are constants.
• Quadratic functions are functions of the form f(x) = ax^2 + bx + c, where a, b and c are constants.
• Cubic functions are functions of the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c and d are constants.
• Note that quadratic and cubic functions exhibit more complex behavior than linear functions.

Inverse Functions

• An inverse function, denoted as f^(-1)(x), is a function that undoes the operation of the original function.
• It flips the x and y in a function.
• Note the difference between the inverse function, written as f^(-1)(x), and a reciprocal function, sometimes described as f^-1(x).

Understanding Domain and Range

• Domain refers to the set of all possible input values (x-values) for a function.
• Range refers to the set of all possible output values (y-values) for a function.
• Understanding how to calculate the domain and the range is crucial when dealing with different calculus equations involving functions.