Functions and Their Representations

Definition and Interpretation of Functions

A function is a relation that assigns each input exactly one output. Functions can be represented in various forms, including tables, graphs, formulas, and verbal descriptions.

• Function notation: denotes the output of function for input .

• Domain: The set of all possible input values () for which the function is defined.

• Range: The set of all possible output values ().

• Example: If is the number of ducks in a lake at year , then is the number of ducks in the initial year.

Function Tables and Graphs

Functions can be represented by tables listing input-output pairs, or by graphs showing the relationship visually.

• Identifying functions from tables: Each input must correspond to exactly one output.

• Piecewise functions: Functions defined by different expressions over different intervals.

• Example:

  x	f(x)
  -2	4
  -1	1
  0	0
  1	1
  2	4

Piecewise Functions

Definition and Construction

A piecewise function is defined by different formulas for different parts of its domain.

• Example: $f(x) = egin{cases} -x^2 & ext{if } x ext{ is odd} \ x^2 & ext{if } x ext{ is even} \\ ext{or} \\ f(x) = egin{cases} -x^2 & x ext{ odd} \\ x^2 & x ext{ even} \\ ext{for } x ext{ in integers} \\ ext{(Additional info: piecewise functions are often used to model situations with abrupt changes.)}

Domain and Range

Finding the Domain

The domain of a function is the set of all input values for which the function is defined.

• For rational functions: Exclude values that make the denominator zero.

• For square root functions: The radicand must be non-negative.

• Example: has domain .

Finding the Range

The range is the set of all possible output values.

• Example: For , the range is .

Average Rate of Change

Definition and Calculation

The average rate of change of a function over the interval is given by:

• Interpretation: Represents the change in output per unit change in input.

• Example: If on , then average rate of change is .

Graphing Functions and Transformations

Basic Graphs and Transformations

Functions can be transformed by shifting, stretching, compressing, or reflecting their graphs.

• Vertical shift: shifts the graph up by units.

• Horizontal shift: shifts the graph right by units.

• Reflection: reflects the graph across the -axis.

• Vertical stretch/compression: stretches if , compresses if .

• Example: shifted up 3 units: .

Piecewise and Transformed Graphs

Piecewise functions and transformations are often graphed to visualize their behavior.

• Example: is shifted right by 1, stretched vertically by 2, and shifted down by 1.

Linear Functions and Equations of Lines

Finding Equations of Lines

The equation of a line in slope-intercept form is , where is the slope and is the -intercept.

• Finding slope:

• Point-slope form:

• Example: The line through and has slope , so .

Applications of Functions

Modeling Real-World Situations

Functions are used to model population growth, rates of change, and other real-world phenomena.

• Example: If a population grows by a fixed amount each year, it can be modeled by a linear function.

• Example: If Terry starts at an elevation of 1,000 ft and descends at 20 ft/sec, his elevation after seconds is .

Classification of Functions: Even, Odd, Neither

Definitions

• Even function: for all in the domain.

• Odd function: for all in the domain.

• Neither: If neither condition holds.

• Example: is even; is odd.

Summary Table: Types of Function Transformations

Additional info:

• Some questions involve interpreting graphs and tables, writing equations for lines, and modeling with functions, all core Precalculus skills.

• Piecewise functions and transformations are emphasized, as are applications to real-world scenarios.