Functions are the backbone of algebra, assigning unique outputs to inputs. They're like vending machines: you put in a value, and it spits out exactly one result. This concept is crucial for modeling real-world relationships and solving complex problems.

In this section, we'll explore function notation, evaluation, and analysis. We'll also dive into different types of functions, their graphs, and key properties. Understanding these concepts will help you tackle more advanced mathematical challenges down the road.

Functions and Function Notation

Functions vs general relations

• Function assigns exactly one output value to each input value
• Relation can have multiple outputs for a single input, functions cannot
• Functions are a subset of relations that satisfy the uniqueness property
  • Each input value is paired with exactly one output value
  • No two ordered pairs in a function can have the same input value with different output values

Evaluation of function inputs

• Function notation: $f(x)$ represents output value of function $f$ when input is $x$
  • If $f(x) = 2x + 1$ and $x = 3$, then $f(3) = 2(3) + 1 = 7$
• Evaluating functions involves substituting given input value for variable in function
  • Simplify expression to find corresponding output value
• Output value represents result of applying function to given input
  • In context, output value can have specific meaning or interpretation (temperature, cost)

One-to-one function analysis

• One-to-one function has each output value corresponding to exactly one input value
  • No two different input values can produce the same output value
• Determining if function is one-to-one involves checking if it passes horizontal line test
  • If any horizontal line intersects graph of function more than once, function is not one-to-one
• One-to-one functions have inverses that are also functions
  • Essential for solving equations and modeling certain real-world situations (encryption, decryption)

Vertical line test for functions

• Graphical method to determine if relation is a function
  • If any vertical line intersects graph more than once, relation is not a function
• Applying vertical line test involves sketching a few vertical lines crossing graph at different points
  • Check if any vertical lines intersect graph more than once
  • If all vertical lines intersect graph at most once, graph represents a function

Graphs of common functions

• Linear functions: $f(x) = mx + b$
  • Graphs as straight lines with slope $m$ and y-intercept $(0, b)$
  • Constant rate of change (slope) between any two points on line
• Quadratic functions: $f(x) = ax^2 + bx + c$
  • Graphs as parabolas, symmetric curves with a vertex (turning point)
  • Shape of parabola depends on value of $a$ (positive or negative)
  • Axis of symmetry and vertex can be found using formula $x = -\frac{b}{2a}$
• Exponential functions: $f(x) = a \cdot b^x$
  • Graphs as curves that increase or decrease rapidly based on value of $b$
    • If $b > 1$, function increases exponentially; if $0 < b < 1$, function decreases exponentially
  • y-intercept is always $(0, a)$, graph approaches x-axis as $x$ approaches negative infinity (assuming $b > 1$)
• Interpreting graphs involves identifying key features (intercepts, vertices, asymptotes)
  • Analyze behavior of function (increasing, decreasing, constant intervals)
  • Relate graphical representation to algebraic form and real-world context, if applicable (population growth, radioactive decay)
• Piecewise functions: defined by different formulas over different intervals of the domain

Function Properties and Operations

• Domain: set of all possible input values for a function
• Range: set of all possible output values of a function
• Codomain: set of all possible output values, including those not actually produced by the function
• Composition: combining two functions to create a new function, denoted as (f ∘ g)(x) = f(g(x))