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Functionsand Their
GraphsChapter 1.2
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Functions
Section 1.2.1
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Relations
Relation: A correspondencebetween 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.
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Relations
Example.
Melissa
John
Jennifer
Patrick
$45,000
$40,000
$50,000
Perso
n
Salary
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Relations
Example.
0
1
4
0
1
{1
2
{2
Numbe
r
Numbe
r
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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 relationthat associates with each element of
X exactly one element of Y
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Functions
Example.Problem: Does this relation represent a
function?
Answer:
Melissa
John
Jennifer
Patrick
$45,000
$40,000
$50,000
Person
Salary
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Functions
Example.Problem: Does this relation represent a
function?
Answer:
0
1
4
0
1
{1
2
{2
Number
Number
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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 ycorresponding to x
Range of the function: the set of allvalues of the function
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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
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Domain and Range
Example. Determine whether therelation represents a function. If it
is a function, state the domain andrange.
Problem:
Relation: f(2,5), (6,3), (8,2), (4,3)g
Answer:
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Domain and Range
Example. Determine whether therelation represents a function. If it
is a function, state the domain andrange.
Problem:
Relation: f(1,7), (0, {3), (2,4), (1,8)g
Answer:
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Equations as Functions
To determine whether an equationis a function
Solve the equation for y. If any value of x in the domain
corresponds to more than one y, theequation doesn¶t define a function
Otherwise, it does define a function.
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Equations as Functions
Example.
Problem: Determine if the equation
x + y2
= 9defines y as a function of x.
Answer:
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Function as a Machine
Accepts numbers from domain asinput.
Exactly one output for each input.
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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:
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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 isgiven explicitly
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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:
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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 thedomain
We usually call x the independentvariable
y is the dependent variable
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Finding the Domain
If the domain isn¶t specified, it willalways be the largest set of real
numbers for which f(x) is a realnumber
We can¶t take square roots of negative numbers (yet) or divide by
zero
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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:
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Finding the Domain
Example. A rectangular gardenhas 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:
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The Graph of aFunction
Section 1.2.2
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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.
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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:
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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:
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Finding Information From
Graphs Example. Answer the
questions about thegraph.
(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
H2, 4¼¼¼¼¼5H- 2, 4¼¼¼¼¼5L H1,2
LH- 1,2
L
0,4
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Finding Information From
Graphs Example. Answer the
questions about thegraph.
(e) Problem: Find thex-intercepts:
Answer:
(f) Problem: Find the
y-intercepts:Answer:
-4 -2 2 4
-4
-2
2
4
H2, 4¼¼¼¼¼5H- 2, 4¼¼¼¼¼5L H1,2
LH- 1,2
L
0,4
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Finding Information From
Graphs Example. Answer the
questions about thegraph.
(g) Problem: How oftendoes 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
H2, 4¼¼¼¼¼5H- 2, 4¼¼¼¼¼5L H1,2
LH- 1,2
L
0,4
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Finding Information From
Formulas Example. Answer the following
questions for the functionf(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 isthe corresponding point on the graph?
Answer:(c) Problem: If f(x) = 1, what is x? What is
(are) the corresponding point(s) on thegraph?
Answer:
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Key Points
Vertical-line Test
Finding Information From Graphs
Finding Information FromFormulas
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Properties of Functions
Section 2.3
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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)
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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)
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Description of Even and
Odd Functions Theorem.
A function is even if and only if its
graph is symmetric with respect tothe y-axis.A function is odd if and only if itsgraph is symmetric with respect to
the origin.
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Description of Even and Odd
Functions Example.
Problem: Doesthe graphrepresent afunctionwhich is
even, odd, or neither?
Answer:
-4 -2 2 4
-4
-2
2
4
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Description of Even and Odd
Functions Example.
Problem: Doesthe graphrepresent afunctionwhich is
even, odd, or neither?
Answer:
-4 -2 2 4
-4
-2
2
4
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Description of Even and Odd
Functions Example.
Problem: Doesthe graphrepresent afunctionwhich is
even, odd, or neither?
Answer:
-4 -2 2 4
-4
-2
2
4
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Identifying Even and OddFunctions 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:
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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 intervalI) For all choices of x in I, the values f(x) are
equal.
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Increasing, Decreasing
and Constant Functions
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Increasing, Decreasing and
Constant Functions Example. Answer the
questions about thefunction shown.
(a) Problem: Where is thefunction increasing?
Answer:
(b) Problem: Where is thefunction decreasing?
Answer:(c) Problem: Where is the
function constant
Answer:
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
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Increasing, Decreasing and
Constant FunctionsWARNING!
Describe thebehavior of a graph
in terms of its x-values.
Answers for thesequestions shouldbe open intervals.
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
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Local Extrema
Local maximum at c:
Open interval I containing x so that, for allx · 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 allx · c in I, f(x) ̧ f(c).
f(c) is a local minimum of f.
Local extrema:
Collection of local maxima and minima
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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.
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Average Rate of Change
Slope of a line can be interpretedas the average rate of change
Average rate of change: If c is in thedomain of y = f(x)
Also called the difference quotient of f at c
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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:
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Secant Lines
Geometric interpretation to theaverage rate of change Label 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 afunction equals the slope of thesecant line containing two points
on its graph
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Secant Lines
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-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:
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Linear
Functions and
ModelsSection 1.2.4
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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 constantaverage rate of change of f(x) = mx + b
is
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-10 -5 5 10
-10
-7.5
-5
-2.5
2.5
5
7.5
10
Linear Functions
Example.
Problem: Graph thelinear function
f(x) = 2x { 5Answer:
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Application: Straight-Line
Depreciation Example. Suppose that a company
has just purchased a new machinefor its manufacturing facility for $120,000. The company choosesto depreciate the machine usingthe straight-line method over 10
years.For straight-line depreciation, thevalue of the asset declines by a
fixed amount every year.
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2 4 6 8 10 12 14
-40000
-20000
20000
40000
60000
80000
100000
120000
140000
Example. (cont.)
(a) Problem: Write a linear functionthat expresses the book value of themachine as a function of its age, x
Answer:
(b) Problem: Graph the linear function
Answer:
Application: Straight-Line
Depreciation
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Example. (cont.)
(c) Problem: What is the book value of the machine after 4 years?
Answer:
(d) Problem: When will the machine beworth $20,000?
Answer:
Application: Straight-Line
Depreciation
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Library of
Functions;Piecewise-defined
FunctionsSection 1.2.5
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Linear Functions
f(x) = mx+b, m and ba 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)
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Constant Function
f(x) = b, b a realnumber
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.
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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.
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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.
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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.
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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
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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
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Reciprocal Function
Domain: x { 0
Range: x { 0
Even/odd/neither : Odd
Decreasing on({1, 0) [ (0, 1)
x-intercepts:None
y-intercept: None
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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
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Absolute Value Function
Can also write the absolute valuefunction as
This is a piecewise-defined
function.
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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
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Greatest Integer Function
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-7.5 -5 -
.5
.5 5 7.5
-8
-6
-4
-
4
6
Piecewise-defined
Functions Example. We candefine a functiondifferently ondifferent 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:
raphing
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raphing
Techniques:Transformation
sSection 2.6
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Transformations
Use basic library of functions andtransformations to plot manyother functions.
Plot graphs that look ³almost´ likeone of the basic functions.
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Shifts
Example.
Problem: Plot f(x) = x3, g(x) = x3 { 1 andh(x) = x3 + 2 on the same axes
Answer:
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
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Shifts
Vertical shift:
A real number k is added to the rightside 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)
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-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Shifts
Example.
Problem: Use thegraph of f(x) = jxj
to obtain thegraph of g(x) = jxj+ 2
Answer:
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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
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Shifts
Horizontal shift:
Argument x of a function f isreplaced 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
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-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Shifts
Example.
Problem: Use thegraph of f(x) = jxj
to obtain thegraph of g(x) = jx+2j
Answer:
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-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Shifts
Example.
Problem: The graphof a function y =
f(x) is given. Use itto plotg(x) = f(x { 3) + 2
Answer:
C i d
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Compressions and
Stretches Example.
Problem: Plot f(x) = x3, g(x) = 2x3 andon the same axes
Answer:
-4 -2 2 4
-4
-3
-2
-1
1
2
3
C i d
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Compressions and
Stretches Vertical compression/stretch:
Right side of function y = f(x) ismultiplied 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)
C i d
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Compressions and
Stretches Example.
Problem: Use thegraph of f(x) = x2
to obtain thegraph of g(x) =3x2
Answer:
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
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Compressions and
Stretches Example.
Problem: Plot f(x) = x3, g(x) = (2x)3
and on the same axesAnswer:
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
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Compressions and
Stretches Horizontal compression/stretch:
Argument x of a function y = f(x) ismultiplied 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)
C i d
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-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Compressions and
Stretches Example.
Problem: Use thegraph of f(x) = x2
to obtain thegraph of g(x) =(3x)2
Answer:
C i d
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-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
Compressions and
Stretches Example.
Problem: The graphof a function y =
f(x) is given. Use itto plotg(x) = 3f(2x)
Answer:
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Reflections
Example.
Problem: f(x) = x3 + 1 and
g(x) = {(x3
+ 1) on the same axesAnswer:
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
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Reflections
Reflections about x-axis :
Right side of the functiony = 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).
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Reflections
Example.
Problem: f(x) = x3 + 1 and
g(x) = ({x)3
+ 1 on the same axesAnswer:
-4 -2 2 4
-4
-3
-2
-1
1
2
3
4
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Reflections
Reflections about y-axis :
Argument of the functiony = 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).
Summary of
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Summary of
Transformations
Summary of
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Summary of
Transformations
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Mathematical
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Mathematical
Models:Constructing
FunctionsSection 1.2.7
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Mathematical Models
Example. Anne has 5000 feet of fencing available to enclose arectangular field. One side of the
field lies along a river, so onlythree 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 tothe river.
Answer:
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1000 2000 3000 4000 5000 6000
500000
1́ 106
1.5´106
2́ 106
2.5´106
3́ 106
3.5´106
Mathematical Models
Example (cont.)
(b) Problem: GraphA = A(x) and find
what value of xmakes the arealargest.
Answer:
(c) Problem: Whatvalue of x makesthe area largest?
Answer: