Functions are the building blocks of math relationships. They show how inputs and outputs connect through equations, tables, graphs, or words. Understanding functions helps us model real-world situations and make predictions.

Functions can be one-to-one or many-to-one. We use tools like the vertical line test to check if a graph represents a function. Common function types include linear, quadratic, and exponential, each with unique shapes and properties.

Functions and Their Representations

Classification of function representations

• Functions represented by equations express the relationship between input and output variables using mathematical symbols and operations
• Functions represented by tables organize input and output values in columns or rows, showing the correspondence between them
• Functions represented by graphs plot input values on the horizontal axis and output values on the vertical axis, visualizing the relationship
• Functions represented by verbal descriptions explain the relationship between input and output using words, often in the context of a real-world situation

Interpretation of function outputs

• Interpreting the output of a function involves understanding what the value represents in the given context
• The units of the output should be considered when interpreting the result (dollars, meters, degrees)
• In a real-world application, the output can provide meaningful information about the situation being modeled (profit, height, temperature)

One-to-one vs many-to-one functions

• One-to-one functions have a unique output for each input, ensuring that no two input values map to the same output value
  • Example: $f(x) = 2x + 1$ is one-to-one because each input produces a unique output
• Many-to-one functions can have multiple input values that map to the same output value
  • Example: $f(x) = x^2$ is many-to-one because both $x = 2$ and $x = -2$ map to the same output value of 4

Vertical line test for functions

• The vertical line test determines if a relation (a set of ordered pairs) is a function based on its graphical representation
• To apply the test, imagine drawing vertical lines through the graph
  • If any vertical line intersects the graph at more than one point, the relation is not a function
  • If no vertical line intersects the graph at more than one point, the relation is a function

Graphs of common functions

• Linear functions have a constant rate of change and are represented by a straight line
  • The slope ($m$) represents the change in $y$ divided by the change in $x$
  • The $y$-intercept ($b$) is the $y$-value when $x = 0$
  • Example: $y = 2x + 3$ is a linear function with a slope of 2 and a $y$-intercept of 3
• Quadratic functions have a parabolic shape and are represented by an equation in the form $y = ax^2 + bx + c$
  • The vertex is the maximum or minimum point of the parabola
  • The axis of symmetry is the vertical line that passes through the vertex
  • Example: $y = -x^2 + 4x - 3$ is a quadratic function with a vertex at $(2, 1)$
• Exponential functions involve a constant base raised to a variable power and are represented by an equation in the form $y = a \cdot b^x$
  • $a$ is the initial value (the $y$-value when $x = 0$)
  • $b$ is the growth factor (if $b > 1$) or decay factor (if $0 < b < 1$)
  • Example: $y = 2 \cdot 3^x$ is an exponential growth function with an initial value of 2 and a growth factor of 3
• Piecewise functions are defined by different equations for different intervals of the input variable

Advanced Function Concepts

• Inverse functions reverse the relationship between input and output, effectively "undoing" the original function
• Composition of functions involves applying one function to the output of another, creating a new function
• Functions can have multiple inputs that together determine a single output