Functions are relationships where one variable depends on another (e.g., y depends on x).
A function must have exactly one output for every input.
Common notation includes y or f(x).
Functions can be represented by tables, graphs, or formulas.
Example of a Function
Fishing Example:
Inputs: Fish caught (fish 1, fish 2, etc.).
Outputs: Weight of each fish.
Each fish has one specific weight, thus it is a function.
Characteristics of Functions
Functions must have exactly one output for each input.
Unique output is not required; outputs can repeat for different inputs.
Graphically, functions can be analyzed using the vertical line test: A graph is a function if no vertical line intersects the graph at more than one point.
Mathematical Representation of Functions
Formulas: E.g., Area of a circle formula, where area depends on the radius.
Graphs: Functions can be represented as graphs, such as y = 3x² - 4x + 2.
Piecewise Functions
Defined by multiple sub-functions, each active over a specific domain.
Example: Absolute value function, defined as x if x >= 0 and -x if x < 0.
Graphing involves plotting each piece within its defined range.
Domain and Range
Domain: All possible input values (x-values).
Range: All possible output values (y-values).
Some functions have restricted domains, e.g., denominators can't be zero.
Finding the Domain
Identify potential problems such as zero denominators or negative values under square roots.
Solve inequalities to find ranges of x that don't lead to undefined expressions.
Odd and Even Functions
Even Functions: Symmetric about the y-axis. Satisfy f(-x) = f(x).
Odd Functions: Symmetric about the origin. Satisfy f(-x) = -f(x).
Examples
Finding Domain: Set the denominator not equal to zero, or ensure radicands are non-negative.
Graphical Analysis: Use vertical and horizontal line tests to determine function characteristics.
Applications
Box-making Problem: Finding a function for the volume of a box made from a piece of cardboard by cutting out squares from the corners and folding up the sides.
Considerations include ensuring cuts lead to a feasible box size.