1.1 Functions and Function Notation

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

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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