Functions An Overview

Functions A function is a procedure for assigning a single output to any acceptable input. Functions can be written as formulas, tables, graphs, or verbal descriptions. Functions are used to describe the relationship between two or more quantities. As the quantities change with respect to each other (cause), they create predictable behaviors (effect) that can be described.

Basic & Elementary Functions Basic functions are power functions, x k (where k is any real number), the trigonometric functions sin(x) and cos(x), and the natural exponential e x and logarithm functions ln (x) An elementary function is a combination of any of these functions through addition, subtraction, multiplication, division, and composition.

Graphs The graph of a function f is the set of points (x, y) that satisfy the equation y = f(x). Graphs may or may not have a formula to represent f(x).

Function Notation Function notation is used to succinctly state the formula for a function and related points. The formula y = 2x written in function notation would appear as f(x) = 2x. Any letter can be chosen as the name of a function, although the letters f, g, and h are the most common.

Function Notation A particular point on the function f(x) = 2x, such as x = 4, is written as f(4) = 8. The corresponding ordered pair notation for this point is (4, 8). The statement y = f(x) is used to indicate that the equation y is also a function in terms of x.

Types of Functions Algebraic functions are combinations of power functions using only addition, subtraction, multiplication and division. Polynomial functions are algebraic functions with the exponents restricted to whole numbers. A rational function is the quotient of two polynomial functions.

Types of Functions A transcendental function is any function which is not algebraic. Exponential, logarithm, and trigonometric functions are all transcendental.

Relations Any set of ordered pairs Can be represented as a  list: {(0, 1); (3, -6); (-7, 2); (3, 5); (0, 2)}  inequality: y > x  equation: x 2 + y 2 = 9  function: y = x 2

Domain and Range Domain:  The set of first entries of the ordered pairs  In a function, the x-values or inputs Range:  The set of second entries of the ordered pairs  In a function, the y-values or outputs

Function A relation that assigns only one range value to any given domain value The formula cannot contain an inequality symbol nor an exponent on the y variable The graph of a function will pass the vertical line test  A vertical line in the xy-plane will intersect the graph of a function in at most one point.

Function Models Functions can model real-life situations in which the values of 2 or more variables are related Many models exist where one variable depends on another  x is the independent variable  y is the dependent variable y is a function of x 

Types of Functions Algebraic  Polynomial  Rational  Root Piecewise Defined  Absolute Value  Greatest Integer Transcendental  Trigonometric  Exponential  Logarithmic Combinations  Sums/Differences  Products/Quotients  Composite

Function Notation is the same as The input variable aka, the “dummy” variable can be replaced by a number, another variable, or an expression is the same as (2, 9)

Identifying Domain & Range y = x  Domain:(-∞, ∞)  Range:(-∞, ∞) y = x 2  Domain:(-∞, ∞)  Range:[0, ∞) y = c(x – a) 2 + b, c < 0  Domain: (-∞, ∞)  Range: (-∞, b] y = x 3  Domain:(-∞, ∞)  Range:(-∞, ∞) y = c(x – a) 2 + b, c > 0  Domain: (-∞, ∞)  Range: [b, ∞) y = x 4  Domain:(-∞, ∞)  Range:[0, ∞)

Algebraic Approach for Range For quadratic functions, “complete the square” to get it in vertex form Range: [4.875, ∞)

Identifying Domain & Range  Domain:[0, ∞)  Range:[0, ∞)  Domain:[a, ∞)  Range:[b, ∞)

Algebraic Approach for Domain For root functions, set the radicand greater than or equal to zero and solve for x Domain: [4/3, ∞)

Identifying Domain & Range  Domain: (-∞, 0)  (0, ∞)  Range: (-∞, 0)  (0, ∞)  Domain: (-∞, a)  (a, ∞)  Range: (-∞, b)  (b, ∞)

Advice Unless otherwise stated, the domain of a relation is taken to be the largest set of real x-values for which there are corresponding real y-values Know the shapes of the basic graphs as well as their domains and ranges

Symmetry of Graphs About the y-axis  If (x, y) is on the graph, so is (-x, y)  Even Functions: f(-x) = f(x) About the x-axis  If (x, y) is on the graph, so is (x, -y)  Not a function About the origin  If (x, y) is on the graph, so is (-x, -y)  Odd Functions: f(-x) = -f(x)

Examples f(x) = x 2  f(-x) = (-x) 2 = x 2  f(-x) = f(x)   even f(x) = x  f(-x) = (-x) = x  f(-x) = f(x)   even f(x) = x  f(-x) = -x  f(-x) = -f(x)   odd f(x) = x + 1  f(-x) = -x+1 = 1-x  f(-x) ≠ f(x) and f(-x) ≠ -f(x)   neither

Piecewise Defined Functions Functions which have different formulas applied to different parts of their domains Example Graphing on calculator – special technique

Absolute Value Function As a piecewise defined function: MATH NUM 1:abs(

Example Draw a complete graph Evaluate algebraically

Integer-Valued Functions Greatest Integer Function  aka, Integer Floor Function   Greatest integer less than or equal to x  MATH NUM 5:int(x)  called a step function  graph in dot mode  Note y-values for negative x-values

Integer-Valued Functions Integer Ceiling Function   Round x up to the nearest integer  Least integer greater than or equal to x

The sum, difference, or product of two functions is also a function  Domain is the intersection of the domains of f and g The quotient of two functions is also a function  Domain is the intersection of the domains of f and g AND  excludes the roots of the denominator function

Composition of Functions Output of 1 function becomes the input to another function Notations   Domain of f  g consists of the range of g which is also in the domain of f

Example