Functions and Their Graphs

Functions and Their Graphs Chapter 2 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAA

Functions Section 2.1

Relations • Relation:A correspondence between two sets. • xcorresponds toy or ydependsonx if a relation exists between x and y • Denote by x ! y in this case.

Relations • Example. Person Salary Melissa John Jennifer Patrick $45,000 $40,000 $50,000

Relations • Example. Number Number 0 1 {1 2 {2 0 1 4

Functions • Function: special kind of relation • Each input corresponds to precisely one output • If X and Y are nonempty sets, a function from X into Y is a relation that associates with each element of X exactly one element of Y

Functions • Example. Problem: Does this relation represent a function? Answer: Person Salary Melissa John Jennifer Patrick $45,000 $40,000 $50,000

Functions • Example. Problem: Does this relation represent a function? Answer: Number Number 0 1 {1 2 {2 0 1 4

Domain and Range • Function from X to Y • Domain of the function: the set X. • If x in X: • The image of x or the value of the function at x: The element y corresponding to x • Range of the function: the set of all values of the function

Domain and Range • Example. Problem: What is the range of this function? Answer: X Y {3 {2 {1 0 1 2 3 0 1 4 9

Domain and Range • Example. Determine whether the relation represents a function. If it is a function, state the domain and range. Problem: Relation: f(2,5), (6,3), (8,2), (4,3)g Answer:

Domain and Range • Example. Determine whether the relation represents a function. If it is a function, state the domain and range. Problem: Relation: f(1,7), (0, {3), (2,4), (1,8)g Answer:

Equations as Functions • To determine whether an equation is a function • Solve the equation for y. • If any value of x in the domain corresponds to more than one y, the equation doesn’t define a function • Otherwise, it does define a function.

Equations as Functions • Example. Problem: Determine if the equation x + y2 = 9 defines y as a function of x. Answer:

Function as a Machine • Accepts numbers from domain as input. • Exactly one output for each input.

Finding Values of a Function • Example. Evaluate each of the following for the function f(x) = {3x2 + 2x (a) Problem:f(3) Answer: (b) Problem:f(x) + f(3) Answer: (c) Problem:f({x) Answer: (d) Problem: {f(x) Answer: (e) Problem:f(x+3) Answer:

Finding Values of a Function • Example. Evaluate the difference quotientof the function Problem:f(x) = { 3x2 + 2x. Answer:

Implicit Form of a Function • A function given in terms of x and y is given implicitly. • If we can solve an equation for y in terms of x, the function is given explicitly

Implicit Form of a Function • Example. Find the explicit form of the implicit function. (a) Problem: 3x + y = 5 Answer: (b) Problem:xy + x = 1 Answer:

Important Facts • For each x in the domain of f, there is exactly one image f(x) in the range • An element in the range can result from more than one x in the domain • We usually call x the independentvariable • y is the dependentvariable

Finding the Domain • If the domain isn’t specified, it will always be the largest set of real numbers for which f(x) is a real number • We can’t take square roots of negative numbers (yet) or divide by zero

Finding the Domain • Example. Find the domain of each of the following functions. (a) Problem:f(x) = x2 { 9 Answer: (b) Problem: Answer: (c) Problem: Answer:

Finding the Domain • Example. A rectangular garden has a perimeter of 100 feet. (a) Problem: Express the area A of the garden as a function of the width w. Answer: (b) Problem: Find the domain of A(w) Answer:

Operations on Functions • Arithmetic on functions f and g • Sum of functions: (f + g)(x) = f(x) + g(x) • Difference of functions: (f {g)(x) = f(x) {g(x) • Domains: Set of all real numbers in the domains of both f and g. • For both sum and difference

Operations on Functions • Arithmetic on functions f and g • Product of functions f and g is (f ¢g)(x) = f(x) ¢g(x) • The quotient of functions f and g is • Domain of product: Set of all real numbers in the domains of both f and g • Domain of quotient: Set of all real numbers in the domains of both f and g with g(x)  0

Operations on Functions • Example. Given f(x) = 2x2 + 3 and g(x) = 4x3 + 1. (a) Problem: Find f+g and its domain Answer: (b) Problem: Find f {g and its domain Answer:

Operations on Functions • Example. Given f(x) = 2x2 + 3 and g(x) = 4x3 + 1. (c) Problem: Find f¢g and its domain Answer: (d) Problem: Find f/g and its domain Answer:

Key Points • Relations • Functions • Domain and Range • Equations as Functions • Function as a Machine • Finding Values of a Function • Implicit Form of a Function • Important Facts • Finding the Domain

Key Points (cont.) • Operations on Functions

The Graph of a Function Section 2.2

Vertical-line Test • Theorem. [Vertical-Line Test]A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graphs in at most one point.

Vertical-line Test • Example. Problem: Is the graph that of a function? Answer:

Finding Information From Graphs • Example. Answer the questions about the graph. (a) Problem: Find f(0) Answer: (b) Problem: Find f(2) Answer: (c) Problem: Find the domain Answer: (d) Problem: Find the range Answer:

Finding Information From Graphs • Example. Answer the questions about the graph. (e) Problem: Find the x-intercepts: Answer: (f) Problem: Find the y-intercepts: Answer:

Finding Information From Graphs • Example. Answer the questions about the graph. (g) Problem: How often does the line y = 3 intersect the graph? Answer: (h) Problem: For what values of x does f(x) = 2? Answer: (i) Problem: For what values of x is f(x) > 0? Answer:

Finding Information From Formulas • Example. Answer the following questions for the function f(x) = 2x2 { 5 (a) Problem: Is the point (2,3) on the graph of y = f(x)? Answer: (b) Problem: If x = {1, what is f(x)? What is the corresponding point on the graph? Answer: (c) Problem: If f(x) = 1, what is x? What is (are) the corresponding point(s) on the graph? Answer:

Key Points • Vertical-line Test • Finding Information From Graphs • Finding Information From Formulas

Properties of Functions Section 2.3

Even and Odd Functions • Even function: • For every number x in its domain, the number {x is also in the domain • f({x) = f(x) • Oddfunction: • For every number x in its domain, the number {x is also in the domain • f({x) = {f(x)

Description of Even and Odd Functions • Even functions: • If (x, y) is on the graph, so is ({x, y) • Odd functions: • If (x, y) is on the graph, so is ({x, {y)

Description of Even and Odd Functions • Theorem. A function is even if and only if its graph is symmetric with respect to the y-axis.A function is odd if and only if its graph is symmetric with respect to the origin.

Description of Even and Odd Functions • Example. Problem: Does the graph represent a function which is even, odd, or neither? Answer:

Identifying Even and Odd Functions from the Equation • Example. Determine whether the following functions are even, odd or neither. (a) Problem: Answer: (b) Problem:g(x) = 3x2 { 4 Answer: (c) Problem: Answer:

Increasing, Decreasing and Constant Functions • Increasing function (on an open interval I): • For any choice of x1 and x2 in I, with x1<x2, we have f(x1) <f(x2) • Decreasing function (on an open interval I) • For any choice of x1 and x2 in I, with x1<x2, we have f(x1) >f(x2) • Constant function (on an open interval I) • For all choices of x in I, the values f(x) are equal.

Increasing, Decreasing and Constant Functions

Increasing, Decreasing and Constant Functions • Example. Answer the questions about the function shown. (a) Problem: Where is the function increasing? Answer: (b) Problem: Where is the function decreasing? Answer: (c) Problem: Where is the function constant Answer:

Increasing, Decreasing and Constant Functions WARNING! • Describe the behavior of a graph in terms of its x-values. • Answers for these questions should be open intervals.

Functions and Their Graphs