Algebra Class: Understanding Functions

Defining Functions

• A function is a relationship between two sets of values, where each input (x-value) is associated with exactly one output (y-value). This relationship can be expressed as an equation, a table, a graph, or a description.
• Functions are often written as f(x) = y, where f(x) represents the output corresponding to the input value x.
• The input values form the domain of the function, and the output values form the range of the function.

Identifying Functions

• A function can be identified in a table if each x-value corresponds to only one y-value.
• On a graph, a vertical line test can be applied. If any vertical line crosses the graph more than once, the graph does not represent a function.

Function Notation

• Function notation, such as f(x) = 2x + 3, allows us to represent the output (y-value) of the function for a particular input (x-value).
• For example, if x = 2, f(2) = 2(2) + 3 = 7.

Types of Functions

• Linear Functions: Functions whose graph is a straight line. These functions can be expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept.
• Quadratic Functions: Functions whose graph is a parabola. These functions can be expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants.
• Exponential Functions: Functions where the input variable is in the exponent. The general form is f(x) = a * b^x, where a and b are constants.

Domain and Range

• The domain of a function comprises all possible input values (x-values).
• The range of a function consists of all possible output values (y-values).
• Determining the domain and range of a function is crucial for understanding its limitations and behavior.

Function Operations

• Adding Functions: (f+g)(x) = f(x) + g(x)
• Subtracting Functions: (f-g)(x) = f(x) - g(x)
• Multiplying Functions: (fg)(x) = f(x) * g(x)
• Dividing Functions: (f/g)(x) = f(x) / g(x), with the additional condition that g(x) ≠ 0.

Composition of Functions

• A composite function combines two functions where the output of one function becomes the input of another function.
• For example, (f∘g)(x) = f(g(x)), meaning the function g(x) is applied first, and then the function f(x) is applied using the result from g(x).

Graphs of Functions

• Understanding graphs allows for visualization of the function's behavior and characteristics.
• Key features to identify on a graph include intercepts, maximum and minimum values, increasing and decreasing intervals, and any asymptotes.

Transformations of Functions

• Transformations like shifting, stretching, or reflecting a function's graph can alter its position and shape without changing its fundamental type.
• Horizontal shifts, vertical shifts, horizontal stretches/compressions, vertical stretches/compressions, and reflections can be identified and analyzed.

Piecewise Functions

• A piecewise function is defined by multiple sub-functions, each applying to a specific interval of the domain.
• Understanding piecewise functions requires attention to the defined intervals for each sub-function.

Inverse Functions

• An inverse function reverses the input-output relationship of a function.
• The inverse of a function f is often denoted as f⁻¹(x).
• The graph of an inverse function is a reflection of the original function across the line y = x.