Relations
Relation: A correspondence between two sets.
x corresponds to y or y
depends on x if a relation
exists between x and y
Denote by x ! y in this case.
Functions
Function: special kind of relationEach 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:
Melissa
John
Jennifer
Patrick
$45,000
$40,000
$50,000
Person
Salary
Functions
Example.Problem: Does this relation represent
a function?Answer:
0
1
4
0
1
{1
2
{2
Number
Number
Domain and Range
Function from X to YDomain 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:
0
1
4
9
{3
{2
{1
0
1
2
3
X Y
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 functionSolve 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:
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 quotient
of 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 independent variable
y is the dependent variable
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 numberWe 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
)()(
)(xgxf
xgf
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
RelationsFunctionsDomain and RangeEquations as FunctionsFunction as a MachineFinding Values of a Function Implicit Form of a Function Important FactsFinding the Domain
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
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Example.Problem: Is the graph that of a
function?Answer:
Vertical-line Test
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
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:
-4 -2 2 4
-4
-2
2
4
2, 452, 4
5 1,21,20,4
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:
-4 -2 2 4
-4
-2
2
4
2, 452, 4
5 1,21,20,4
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:
-4 -2 2 4
-4
-2
2
4
2, 452, 4
5 1,21,20,4
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:
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)Odd function:
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:
-4 -2 2 4
-4
-2
2
4
Description of Even and Odd Functions
Example. Problem:
Does the graph represent a function which is even, odd, or neither?
Answer:
-4 -2 2 4
-4
-2
2
4
Description of Even and Odd Functions
Example. Problem:
Does the graph represent a function which is even, odd, or neither?
Answer:
-4 -2 2 4
-4
-2
2
4
Identifying Even and Odd Functions from
the EquationExample. 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
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:
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Increasing, Decreasing and Constant FunctionsWARNING!
Describe the behavior of a graph in terms of its x-values.
Answers for these questions should be open intervals.
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Local Extrema Local maximum at c:
Open interval I containing x so that, for all x · c in I, f(x) · f(c).
f(c) is a local maximum of f. Local minimum at c:
Open interval I containing x so that, for all x · c in I, f(x) ¸ f(c).
f(c) is a local minimum of f. Local extrema:
Collection of local maxima and minima
Local Extrema
For local maxima:Graph is increasing to the left
of c Graph is decreasing to the right
of c. For local minima:
Graph is decreasing to the left of c
Graph is increasing to the right of c.
Local Extrema Example. Answer
the questions about the given graph of f.
(a) Problem: At which
number(s) does f
have a local
maximum?
Answer:
(b) Problem: At which
number(s) does f
have a local
minimum?
Answer:
-7.5 -5 -2.5 2.5 5 7.5
-6
-4
-2
2
4
6
Average Rate of Change
Slope of a line can be interpreted as the average rate of changeAverage rate of change: If c is
in the domain of y = f(x)
Also called the difference quotient of f at c
Average Rate of Change
Example. Find the average rates of change of
(a) Problem: From 0 to 1.
Answer:
(b) Problem: From 0 to 3.
Answer:
(c) Problem: From 1 to 3:
Answer:
Secant LinesGeometric interpretation to the
average rate of changeLabel two points (c, f(c)) and (x,
f(x))Draw a line containing the points. This is the secant line.
Theorem. [Slope of the Secant Line]The average rate of change of a function equals the slope of the secant line containing two points on its graph
-7.5 -5 -2.5 2.5 5 7.5
-5
-2.5
2.5
5
7.5
10
12.5
15
Secant Lines Example.
Problem: Find an
equation of the
secant line to
containing (0,
f(0)) and (5,
f(5))
Answer:
Key Points
Even and Odd FunctionsDescription of Even and Odd
Functions Identifying Even and Odd
Functions from the Equation Increasing, Decreasing and
Constant FunctionsLocal ExtremaAverage Rate of Change
Linear Functions
Linear function: Function of the form f(x) = mx + b Graph: Line with slope m and y-
intercept b.
Theorem. [Average Rate of Change of Linear Function]Linear functions have a constant average rate of change. The constant average rate of change of f(x) = mx + b is
-10 -5 5 10
-10
-7.5
-5
-2.5
2.5
5
7.5
10
Linear Functions
Example. Problem: Graph
the linear functionf(x) = 2x { 5
Answer:
Application: Straight-Line Depreciation
Example. Suppose that a company has just purchased a new machine for its manufacturing facility for $120,000. The company chooses to depreciate the machine using the straight-line method over 10 years.For straight-line depreciation, the value of the asset declines by a fixed amount every year.
2 4 6 8 10 12 14
-40000
-20000
20000
40000
60000
80000
100000
120000
140000
Example. (cont.)(a) Problem: Write a linear
function that expresses the book value of the machine as a function of its age, x
Answer: (b) Problem: Graph the linear
function Answer:
Application: Straight-Line Depreciation
Example. (cont.)(c) Problem: What is the book
value of the machine after 4 years?
Answer: (d) Problem: When will the
machine be worth $20,000? Answer:
Application: Straight-Line Depreciation
Scatter Diagrams
Example. The amount of money that a lending institution will allow you to borrow mainly depends on the interest rate and your annual income.
The following data represent the annual income, I, required by a bank in order to lend L dollars at an interest rate of 7.5% for 30 years.
Scatter Diagrams
Example. (cont.) Annual Income, I ($)
Loan Amount, L ($)
15,000 44,600
20,000 59,500
25,000 74,500
30,000 89,400
35,000 104,300
40,000 119,200
45,000 134,100
50,000 149,000
55,000 163,900
60,000 178,800
65,000 193,700
70,000 208,600
Scatter Diagrams
Example. (cont.) Problem: Use a graphing utility
to draw a scatter diagram of the data.
Answer:
Linear and Nonlinear Relationships
0 .0
1 .0
2 .0
3 .0
4 .0
5 .0
6 .0
7 .0
-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5
0 .0
1 .0
2 .0
3 .0
4 .0
5 .0
6 .0
7 .0
8 .0
9 .0
1 0 .0
-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5
-1 0 .0
-8 .0
-6 .0
-4 .0
-2 .0
0 .0
2 .0
4 .0
6 .0
8 .0
1 0 .0
-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5
-1 2 .0
-1 0 .0
-8 .0
-6 .0
-4 .0
-2 .0
0 .0
2 .0
4 .0
6 .0
8 .0
-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5
-1 5 .0
-1 0 .0
-5 .0
0 .0
5 .0
1 0 .0
1 5 .0
-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5
-1 5 .0
-1 0 .0
-5 .0
0 .0
5 .0
1 0 .0
1 5 .0
-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5
Linear Nonlinear
Linear
Nonlinear Linear Nonlinear
Line of Best Fit
For linearly related scatter diagramLine is line of best fit.Use graphing calculator to find
Example.(a) Problem: Use a graphing
utility to find the line of best fit to the data in the last example.
Answer:
Line of Best Fit
Example. (cont.) (b) Problem: Graph the line of
best fit from the last example on the scatter diagram.
Answer:
Line of Best Fit
Example. (cont.)(c) Problem: Determine the loan
amount that an individual would qualify for if her income is $42,000.
Answer:
Direct Variation
Variation or proportionality. y varies directly with x, or is
directly proportional to x: There is a nonzero number
such that y = kx.
k is the constant of proportionality.
Direct Variation
Example. Suppose y varies directly with x. Suppose as well that y = 15 when x = 3. (a) Problem: Find the constant
of proportionality. Answer:
(b) Problem: Find x when y = 124.53.
Answer:
Key Points
Linear FunctionsApplication: Straight-Line
DepreciationScatter DiagramsLinear and Nonlinear
RelationshipsLine of Best FitDirect Variation
Linear Functions f(x) = mx+b, m and
b a real number Domain: ({1, 1)
Range: ({1, 1)
unless m = 0
Increasing on ({1, 1)
(if m > 0)
Decreasing on ({1, 1)
(if m < 0)
Constant on ({1, 1)
(if m = 0)
Constant Function f(x) = b, b a real
number Special linear
functions Domain: ({1, 1) Range: fbg Even/odd/neither:
Even (also odd if b = 0)
Constant on ({1, 1) x-intercepts: None
(unless b = 0) y-intercept: y = b.
Identity Function f(x) = x
Special linear function
Domain: ({1, 1) Range: ({1, 1) Even/odd/neither:
Odd Increasing on ({1,
1) x-intercepts: x =
0 y-intercept: y = 0.
Square Function f(x) = x2
Domain: ({1, 1) Range: [0, 1) Even/odd/neither:
Even Increasing on (0,
1) Decreasing on
({1, 0) x-intercepts: x =
0 y-intercept: y = 0.
Cube Function
f(x) = x3
Domain: ({1, 1) Range: ({1, 1) Even/odd/
neither: Odd Increasing on
({1, 1) x-intercepts: x =
0 y-intercept: y =
0.
Square Root Function
Domain: [0, 1) Range: [0, 1) Even/odd/
neither: Neither Increasing on
(0, 1) x-intercepts: x
= 0 y-intercept: y =
0
Cube Root Function
Domain: ({1, 1) Range: ({1, 1) Even/odd/neither:
Odd Increasing on ({1,
1) x-intercepts: x = 0 y-intercept: y = 0
Reciprocal Function
Domain: x 0 Range: x 0 Even/odd/
neither: Odd Decreasing on
({1, 0) [ (0, 1) x-intercepts:
None y-intercept:
None
Absolute Value Function
f(x) = jxj Domain: ({1, 1) Range: [0, 1) Even/odd/neither:
Even Increasing on (0,
1) Decreasing on
({1, 0) x-intercepts: x =
0 y-intercept: y = 0
Absolute Value Function
Can also write the absolute value function as
This is a piecewise-defined function.
Greatest Integer Function
f(x) = int(x) greatest integer
less than or equal to x
Domain: ({1, 1) Range: Integers
(Z) Even/odd/
neither: Neither y-intercept: y =
0 Called a step
function
-7.5 -5 -2.5 2.5 5 7.5
-8
-6
-4
-2
2
4
6
8
Piecewise-defined Functions
Example. We can define a function differently on different parts of its domain.(a) Problem: Find
f({2) Answer:
(b) Problem: Find f({1) Answer:
(c) Problem: Find f(2) Answer:
(d) Problem: Find f(3) Answer:
Key Points
Linear FunctionsConstant Function Identity FunctionSquare FunctionCube FunctionSquare Root FunctionCube Root FunctionReciprocal FunctionAbsolute Value Function
Transformations
Use basic library of functions and transformations to plot many other functions.
Plot graphs that look “almost” like one of the basic functions.
Shifts
Example. Problem: Plot f(x) = x3, g(x) = x3
{ 1 and h(x) = x3 + 2 on the same axes
Answer:
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Shifts
Vertical shift:A real number k is added to
the right side of a function y = f(x),
New function y = f(x) + k Graph of new function:
Graph of f shifted vertically up k units (if k > 0)
Down jkj units (if k < 0)
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Shifts
Example. Problem: Use
the graph of f(x) = jxj to obtain the graph of g(x) = jxj + 2
Answer:
Shifts
Example. Problem: Plot f(x) = x3, g(x) = (x
{ 1)3 and h(x) = (x + 2)3 on the same axes
Answer:
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Shifts
Horizontal shift:Argument x of a function f is
replaced by x { h, New function y = f(x { h) Graph of new function:
Graph of f shifted horizontally right h units (if h > 0)
Left jhj units (if h < 0)Also y = f(x + h) in latter case
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Shifts
Example. Problem: Use
the graph of f(x) = jxj to obtain the graph of g(x) = jx+2j
Answer:
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Shifts
Example. Problem: The
graph of a function y = f(x) is given. Use it to plot g(x) = f(x { 3) + 2
Answer:
Compressions and Stretches
Example. Problem: Plot f(x) = x3, g(x) =
2x3 and on the same axes
Answer:
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Compressions and Stretches
Vertical compression/stretch:Right side of function y = f(x) is
multiplied by a positive number a,
New function y = af(x)Graph of new function:
Multiply each y-coordinate on the graph of y = f(x) by a.
Vertically compressed (if 0 < a < 1)Vertically stretched (if a > 1)
Compressions and Stretches
Example. Problem: Use
the graph of f(x) = x2 to obtain the graph of g(x) = 3x2
Answer:
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Compressions and Stretches
Example.
Problem: Plot f(x) = x3, g(x) =
(2x)3
and on the same
axes
Answer:-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Compressions and Stretches
Horizontal compression/stretch:Argument x of a function y = f(x)
is multiplied by a positive number a
New function y = f(ax)Graph of new function:
Divide each x-coordinate on the graph of y = f(x) by a.
Horizontally compressed (if a > 1)Horizontally stretched (if 0 < a < 1)
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Compressions and Stretches
Example. Problem: Use
the graph of f(x) = x2 to obtain the graph of g(x) = (3x)2
Answer:
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Compressions and Stretches
Example. Problem: The
graph of a function y = f(x) is given. Use it to plot g(x) = 3f(2x)
Answer:
Reflections
Example.
Problem: f(x) = x3 + 1 and
g(x) = {(x3 + 1) on the same
axes
Answer:
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Reflections
Reflections about x-axis :Right side of the function
y = f(x) is multiplied by {1, New function y = {f(x)Graph of new function:
Reflection about the x-axis of the graph of the function y = f(x).
Reflections
Example.
Problem: f(x) = x3 + 1 and
g(x) = ({x)3 + 1 on the same
axes
Answer:
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Reflections
Reflections about y-axis :Argument of the function
y = f(x) is multiplied by {1, New function y = f({x)Graph of new function:
Reflection about the y-axis of the graph of the function y = f(x).
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Summary of Transformations
Example. Problem: Use transformations to
graph the functionAnswer:
Mathematical Models
Example.
Problem: The volume V of a
right circular cylinder is V =
¼r2h. If the height is three
times the radius, express the
volume V as a function of r.
Answer:
Mathematical Models
Example. Anne has 5000 feet of fencing available to enclose a rectangular field. One side of the field lies along a river, so only three sides require fencing.(a) Problem: Express the area A
of the rectangle as a function of x, where x is the length of the side parallel to the river. Answer:
1000 2000 3000 4000 5000 6000
500000
1106
1.5106
2106
2.5106
3106
3.5106
Mathematical Models
Example (cont.)(b) Problem:
Graph A = A(x) and find what value of x makes the area largest.
Answer:(c) Problem: What
value of x makes the area largest? Answer: