Math: Introduction to Functions

Introduction to Functions

Functions vs. Relations

• A function is a relation in which each input has only one output.
• Not all relations are functions.

Identifying Functions

• A real-world action can be a function, but not all real-world actions are functions.
• Examples of real-world actions that are not functions: throwing a ball, playing a musical instrument.
• Examples of real-world actions that are functions: baking a cake, measuring the length of an object.

Table Representation

• A table can be used to represent a function, where each input corresponds to exactly one output.
• Example: given a table, find the output value for a given input, e.g., find f(x) when x = 2.
• The inverse of a function, denoted by f^(-1), can be found by swapping the input and output values in the table.

Graphical Representation

• A graph can be used to represent a function, where each input corresponds to exactly one output.
• A vertical line test can be used to determine if a graph represents a function: if a vertical line intersects the graph at more than one point, it is not a function.
• Example: identify which graph represents a function.

Domain and Range

• The domain of a function is the set of input values, often represented by x.
• The range of a function is the set of output values, often represented by y.
• Example: find the domain and range of a given graph or function.

Transformations of Functions

• Functions can be transformed by applying vertical or horizontal shifts, reflections, or stretches.
• Example: solve for a transformed function, e.g., f(x) = 2(x + 1) - 3.

Composite Functions

• A composite function is a function that takes another function as its input.
• Example: given two functions, find the composite function, e.g., if f(x) = 2x and g(x) = x + 1, find (f ∘ g)(x).

More Questions

• If f(x) = x^2, what is the value of f(-2)?
• If g(x) = 2x - 1, what is the value of g(3)?
• If h(x) = x^2 + 1, what is the value of h(-1)?