Welcome to Precalculus

1
Welcome to Precalculus

• Although there will not be a lecture over
  Appendix A.1, it will be assumed that you are
  familiar with the material covered there. Make
  sure you read and understand Appendix A.1.
• Maple software will often be used in class to
  plot graphs. You will not be responsible for
  learning Maple unless you want to. Maple is
  available on computers in math labs on campus.

2
Pythagoras' Theorem

• For a right triangle with hypotenuse of length c
  and sides of lengths a and b, it is true that a2
  b2 c2. The converse is also true. That is,
  if a2 b2 c2 for some labeling of the sides of
  a triangle, then the triangle is a right triangle.

c
a
b
3
The Distance Formula

• The distance d between the points (x1, y1) and
  (x2, y2) in the plane is
• Example. Show that the triangle with vertices
  (2,3), (3,2), and (6, 1) is a right
  triangle.
• If we can verify that the sum of the squares of
  the sides equals the square of the hypotenuse,
  then this is a right triangle, and it follows
  since

4
The Midpoint Formula

• The coordinates (xm, ym) of the midpoint M of the
  line segment with endpoints (x1, y1) and (x2, y2)
  are given by
• Problem. Find the coordinates of the midpoint of
  the line segment with endpoints (2,1) and (2,4).
  
  Solution. The midpoint is (2, 5/2).

5
Translating points in the plane

• A square is shown in the figure. Find the
  coordinates of the vertices of the square shifted
  2 units to the right and 1 unit downward.

y
Original Square
Translated Square
x
Original vertex (x, y). Translated vertex (x2,
y1).
6
Sketching a graph

• Sketch the graph of x2 y x 0.
• First, we solve for y y x2 x.
• Next, we make a table of values.
• Next, we plot the points and connect them with a
  smooth line.

x 2 1 0 1 2 3
y x2x 6 2 0 0 2 6
(x, y) (2, 6) (1,2) (0, 0) (1, 0) (2,2) (3, 6)
?
?
?
?
?
?
7
Intercepts of a graph

• ? A point at which a graph has 0 as the
  y-coordinate is called an x-intercept. Likewise,
  a point at which a graph has 0 as the
  x-coordinate is called a y-intercept. Sometimes
  we will refer to the nonzero coordinate as the
  x-intercept, y-intercept, resp.
• ? To find x-intercepts, set y to zero and
  solve for x. To find y-intercepts, set x
  to zero and solve for y.
• Example. The graph of y (x2)21 is shown
  below.

From the graph, we see that the x-intercepts are
(1,0) and (3,0), and the y-intercept
is (0,3).
?
?
?
8
Algebraic Tests for Symmetry

• The graph of an equation is symmetric with
  respect to the (i) x-axis if
  replacing y with y results in an equivalent
  equation. (ii) y-axis
  if replacing x with x results in an equivalent
  equation. (iii) origin if
  replacing x with x and y with y results in an
  equivalent equation.
• Example. Test the equation xy 5 for symmetry.
  Since (x)(y) 5 is not
  equivalent, it is not symmetric wrt
  x-axis. Since (x)(y)
  5 is not equivalent, it is not symmetric wrt
  y-axis. Since (x)(y) 5 is
  equivalent, it is symmetric wrt the origin.

9
Circles

• A point (x, y) is on a circle with center (h, k)
  if and only if its distance from the center is
  equal to the radius r. By the distance formula,
• If we square both sides of the latter equation,
  we get the standard form of the equation of a
  circle with center (h, k) and radius r.

If h k 0 and r 1, we get the equation of
the unit circle which will be important later
when we study trigonometric functions.
10
The slope-intercept form of a linear equation

• The slope-intercept form is
  where m is the slope and b is the
  y-intercept.
• The y-intercept, b, tells us where the line
  crosses the y-axis. If the slope, m, is positive,
  the line climbs from left to right. If the slope,
  m, is negative, the line falls from left to
  right.
• The larger the value of m (either positive or
  negative), the steeper the graph of the linear
  equation y mx b.
• If (x0, y0) and (x1, y1) are two distinct points
  on the graph of a line, then

11
Slope as a rate of change

• The slope of a nonvertical line is the number of
  units the line rises (or falls) vertically for
  each unit of horizontal change from left to right.

y
m units, mgt0
1 unit
x
y
1 unit
m units, mlt0
x
12
Finding a linear equation from a table

• Suppose the value of a Batman comic book is
  increasing as follows
• If we let year 2000 correspond to t 0, the
  table becomes
• Since
  is constant,
  we let m
  and the equation is

Year 2000 2002 2004 2006
Value () 65 90 115 140
t (year) 0 2 4 6
v () 65 90 115 140
13
Usage of duplicating paper at Lee High School

• At present there are 400 packages of duplicating
  paper available. Each week 12 packages are used.
  A table is shown next which gives the number of
  packages left, L, versus the number of weeks from
  now, w.
• The relation between L and w is linear. Such a
  linear relation can be given a formula of the
  type where m is the slope
  or rate of change and b is the vertical
  intercept. Can you tell what the values of m and
  b are in this situation? What is the
  significance of the horizontal intercept?

w 1 2 3 4
5 6 7
L 388 376 364 352 340
328 316
14
Other equations of lines

• The point-slope form is
  where m is the slope and (x0, y0) is a point on
  the line.
• The general form is
  where A, B, and C are constants.

15
Useful facts about equations of lines

• For any constant k The graph of the
  equation y k is a horizontal line and its
  slope is zero. The graph of the
  equation x k is a vertical line and its slope
  is undefined.
• Let L1 and L2 be two lines having slopes m1 and
  m2, respectively. Then These lines are
  parallel if and only if m1 m2. These
  lines are perpendicular if and only if

16
Use of Maple to plot perpendicular lines
gt plot(2x,(1/2)x5, x0..4,3..5,
colorblack, scalingconstrained)
17

• Example. Use the point-slope form of the line to
  derive the equation which converts temperature in
  degrees Celsius, C, to degrees Fahrenheit, F.
  We are given that the slope is 9/5 and that
  (20,68) is a point on the line. That is, C0 20
  and F0 68. Using the point-slope form, we
  have
• The equation may be rewritten in slope-intercept
  form as

18
Use of Maple to graph temperature conversion
gt plot((9/5)C32,C0..40,colorblack,labels"C",
"F")
19
Common formulas for Area A, Perimeter P, and
Circumference C

• For a rectangle A lw and P 2l 2w
• For a circle A ?r2 and C 2?r

w
l
r
20
Introduction to functions

• When two quantities are related to each other by
  some rule of correspondence, we call the
  correspondence a relation.
• A function f from a set A to a set B is a
  relation that assigns to each element x from A
  exactly one element y from the set B. The set A
  is the domain (or set of inputs) of f, and the
  set B contains the range (or set of outputs).
• Example. A certain childs height (in inches) on
  his birthday is a function of his age (in years)

Age 1 2 3 4 5
Height 24 30 37 41 48
21
Understanding the function definition

• This represents a function
• This does not represent a function from A to B.
  (Why?)

?
1
1
?
?
3
?
2
?
5
Range 1, 5 is subset of B
Domain 1, 2 A
?
1
1
?
?
3
2
?
?
5
A
B
22
Testing for functions represented algebraically

• Which of these equations represents y as a
  function of x?
• Solution. a. If x is known, y x2 is
  determined uniquely, so this is a
  function. b. If x is known, there
  are two values for y, namely, so this
  is not a function.

23
Four ways to represent a function

• Each batch of 5 dozen sugar cookies requires 2
  and one-half cups of flour (verbally).
• We may also use a table (numerically).
• Graphically
• Algebraically F 2.5B

No. of Batches (B) 1
2 3
No. of Cups Flour (F) 2.5
5.0 7.5
F 7.5 5.0 2.5
1 2 3 B
24

• To indicate that a quantity y is a function of a
  quantity x, we abbreviate to y equals f
  of x and, using function notation,
  to
• Here, y is the dependent (or output) variable and
  x is the independent (or input) variable.
• In the previous cookie example, F is the
  dependent variable and B is the independent
  variable, and we can write F
  f(B) 2.5B Note that we could use
  another letter instead of f. How about c for
  cookie? Then F c(B) 2.5B

25

• Example of quantities which are related, but
  neither quantity is a function of the other.
• Both F and R are functions of t. However, F is
  not a function of R, and R is not a function of F
  (do you see why?). In other words, if we know
  which month is being discussed, we can determine
  the values of F and R uniquely. However, if we
  only know the value of F, then the value of R may
  not be determined uniquely. Similarly, if we
  only know the value of R, then F may not be
  uniquely determined.

t, month 1 2 3 4
5 6 7
R, no.rabbits 750 567 500 567 750 1000
1250
F, no.foxes 143 125 100 75 57
50 57
26
Evaluating a function

• Evaluating a function means figuring out the
  value of a functions output from a particular
  value of the input.
• Example. Let the function g be defined
  by

27
Evaluating functions using a table

• Suppose that f is defined by the table
• To find f(3), we look in the table and get f(3)
  4.
• Now define g(x) f(x1). We evaluate g(3)
  f(4) 3. The table for g appears below
• Why is no value listed for g(4)? Why is g
  defined at x 2 while f is not?

x 1 0 1
2 3 4
f(x) 2 1 5
2 4 3
x 2 1 0
1 2 3 4
g(x) 2 1 5
2 4 3 --
28

• Given an input, we evaluate a function to find
  the output. Often the situation is reversed we
  know the output value and we want to find the
  corresponding input value(s). If the function is
  given by a formula, the input values are
  solutions to an equation.
• Problem. Let A f(r) be the area of a circle of
  radius r, where r is in cm. What is the radius
  of a circle whose area is 100 cm2 ?
  Solution. The output f(r) is an area. Solving
  the equation f(r) 100 for r gives us the radius
  of a circle whose area is 100 cm2. Since the
  formula for the area of a circle is
  we solve
  This yields and
  we take the positive value.

29
Finding input and output values using a table

• Suppose that f is defined by the table
• As before, to find f(1), we look in the table and
  get f(1) 5.
• Now suppose we want to solve f(x) 2 for x.
• There are two values for x which satisfy
  this condition,
• namely, x 1 and x 2.

x 1 0 1
2 3 4
f(x) 2 1 5
2 4 3
30
Input and output from a graph showing an
influenza epidemic

• We have I f(w), where I is the number of
  individuals infected (in thousands) w weeks after
  the epidemic begins.
• Evaluate f(2) and explain its meaning.
• Solve f(w) 4.5 and explain the meaning of the
  solution.

31
More on input and output from the graph of a
function

• Use the graph of f to the right to find
  or estimate u
• f(4) u
• Is u an input or an output?
• f(f(8)) u
• Is u an input or an output?
• f(u) 24
• Is u an input or an output?

32
Evaluating a difference quotient--important in
calculus

• For a given function f, the ratio
  is called a
  difference quotient.
• Evaluate the difference quotient for f(x) x2.

33
Evaluating a piecewise defined function

• Evaluate the function f at x 2, x 0, x 3.
• Solution.

34
Suppose an employee is paid 5.00 per hour to
work a standard 40 hour work week. If he works
overtime, he is paid 7.50 per hour up to a
maximum of 80 hours. The graph below shows his
weekly pay as a function, f(t), of the time
worked.
p2
?
?
p1
What are the values of p1 and p2?
35
Pay function in bracket form
(80,500)
?
?
(40,200)
f(t)
36
Domain of a function

• The domain of a function can be described
  explicitly
  Domain 1, 2, 3, 4
• The domain may be determined from context
  Area of a circle as a function of the
  radius is g(r) ?r2, and the domain is all real
  numbers r gt 0.
• The domain may be implied. That is, it is all
  real numbers for which the expression involved is
  defined. For example

x 1 2 3 4
f(x) 5 7 2 3
37
The graph of a function

• The graph of a function f is the collection of
  ordered pairs (x, f(x)) such that x is in the
  domain of f.
• The graph of f can be visualized by plotting it
  in a coordinate plane.
• Example. f(x) x3x has the graph shown below.

38
Finding the domain and range for a function whose
graph is given
?
Range
?
Domain
Domain 1.5, 5.5) Range 1,3
39
Vertical Line Test for functions

• A set of points in a coordinate plane is the
  graph of y as a function of x if and only if no
  vertical line intersects the graph at more than
  one point.
• In which of the graphs below could y be a
  function of x?
• Clearly, the one on the right fails the vertical
  line test, so this graph does not represent y as
  a function of x.

y
y
x
x
40
Zeros of a function

• The zeros of a function f of x are the x-values
  for which f(x) 0.
• In terms of the graph of f, the zeros are the
  x-coordinates of the points where the graph
  crosses the x-axis.
• To find the zeros of f, set f(x) 0 and solve
  for x.
• Example. Find the zeros of f(x) x2 x 2.

41
Increasing, decreasing, and constant functions

• A function f is increasing on an interval if for
  any x1 and x2 in the interval,
• A function f is decreasing on an interval if for
  any x1 and x2 in the interval,
• A function f is constant on an interval if for
  any x1 and x2 in the interval,

Decreasing on ( ?,0)
Increasing on (2,?)
Constant on (0,2)
42
Relative minimum and relative maximum

• A function value f(a) is called a relative
  minimum of f if there exists an interval (x1, x2)
  that contains a such that
• A function value f(a) is called a relative
  maximum of f if there exists an interval (x1, x2)
  that contains a such that

y
Relative maximum
?
x
?
Relative minima
?
43
Average rate of change

• For a function f, the average rate of change
  between two points
  is the slope of the line through the two
  points. This line is called the secant line and
  its slope is denoted

Secant line
?
?
44
Tests for even and odd functions

• A function is called an even function if, for all
  values of x in the domain of f,
  The graph of an even function is symmetric across
  the y-axis. Examples of even functions are power
  functions with even exponents, such as y x2, y
  x4, y x6, ...
• A function is called an odd function if, for all
  values of x in the domain of f,
  The graph of an odd function is symmetric about
  the origin. Examples of odd functions are power
  functions with odd exponents, such as y x1, y
  x3, y x5, ...

45

• Problem. Is the function f(x) x3x even, odd,
  or neither? Solution. Since
  2 f(1) is not equal to f(1) 2, it follows
  that f is not even. Since
  f(x)
  f(x), it follows that f is odd.

y x3x
Note the symmetry about the origin.
46

• Problem. Is the function f(x) x even, odd,
  or neither? Solution. Since
  f(x) x f(x), it follows that f is
  even. Since 1 f(1) is not equal to
  f(1) 1, it follows that f is not odd.
• Question. Is it possible for a function to be
  both even and odd?

y x
Note the symmetry about the y-axis.
47
A library of parent functions

• Next, eight functions which are the most commonly
  used functions in algebra will be presented.
• These functions will be called parent functions
  because they can be used to create many other
  functions using transformations which will be
  introduced later.
• Memorize the names and characteristics of these
  functions.

48
The constant function

• Domain is all real numbers
• Range is c
• The function is even for all c.
• The function is odd when c 0.

49
The identity function

• Domain is all real numbers
• Range is all real numbers
• The function is odd

50
The absolute value function

• Domain is all real numbers
• Range is all nonnegative real numbers
• The function is even

51
The squaring function

• Domain is all real numbers
• Range is all nonnegative real numbers
• The function is even

52
The cubic function

• Domain is all real numbers
• Range is all real numbers
• The function is odd

53
The square root function
?

• Domain is all nonnegative real numbers
• Range is all nonnegative real numbers
• The function is neither even nor odd

54
The reciprocal function

• Domain is all real numbers except 0
• Range is all real numbers except 0
• The function is odd

55
The greatest integer function
?
?
?
?
?
?
?

• Domain is all real numbers
• Range is all integers
• The function is neither even nor odd

56
Name the functions
(i)
(ii)
(iii)
(iv)
57
Name the functions
(v)
(vi)
?
?
?
?
?
?
?
(viii)
(vii)
58
Graphing a piece-wise defined function

• Sketch the graph of

?
59
Transformation of functions--vertical and
horizontal shifts
Let c be a positive real number. Vertical and
horizontal shifts in the graph of y f(x) are
represented as follows.

1. Vertical shift c units upward h(x)
   f(x) c
2. Vertical shift c units downward h(x)
   f(x) c
3. Horizontal shift c units to the right h(x)
   f(x c)
4. Horizontal shift c units to the left h(x)
   f(x c)

60
Example showing that shifts can be combined
The squaring function is shifted right by 2 units
and upward by 1 unit.
61
Transformation of functions--reflections
Reflections in the coordinate axes of the graph
of y f(x) are represented as follows. 1.
Reflection in the x-axis h(x) f(x) 2.
Reflection in the y-axis h(x) f(x)
?
?
62
Example showing that reflections and shifts can
be combined
?
?
Note that domain of h(x) is
63
Transformation of functions--nonrigid
transformations

• Horizontal shifts, vertical shifts, and
  reflections are rigid transformations because the
  basic shape of the graph is unchanged. Only the
  position of the graph is transformed.
• Nonrigid transformations, which are introduced
  next, cause a change in the shape of the original
  graph.

64
Transformation of functions--vertical stretch and
vertical shrink

• Vertical nonrigid transformations of the
  graph of y f(x) are represented as follows.
• Vertical stretch h(x) cf(x), c gt 1
• 2. Vertical shrink h(x) cf(x), 0 lt c lt 1

65
Transformation of fcts--horizontal stretch and
horizontal shrink

• Horizontal nonrigid transformations of the
  graph of y f(x) are represented as follows.
• Horizontal shrink h(x) f(cx), c gt 1
• Horizontal stretch h(x) f(cx), 0 lt c lt 1
• See the next slide for examples.

66

• Let f(x) 4x2, h1(x) 4 (2x)2, and h2(x) 4
  (0.5x)2. Then h1 is a horizontal shrink of f
  and h2 is a horizontal stretch of f.

67
Arithmetic combination of functions--sum function
Let f and g be two functions with overlapping
domains. Then, for all x common to both domains,
the sum of f and g is defined as follows.
Example. f(x) x2 2 g(x)
x2 1 (fg)(x) 2x23
68
Arithmetic combination of functions--difference
function
Let f and g be two functions with overlapping
domains. Then, for all x common to both domains,
the difference of f and g is defined as follows.
Example. f(x) x2 2 g(x)
x2 1 (f g)(x) 1
69
Arithmetic combination of functions--product
function
Let f and g be two functions with overlapping
domains. Then, for all x common to both domains,
the product of f and g is defined as follows.
Example. f(x) x2 2 g(x)
x2 1 (fg)(x) (x2 2)(x2 1)
x4 3x2 2
70
Arithmetic combination of functions--quotient
function
Let f and g be two functions with overlapping
domains. Then, for all x common to both domains,
the quotient of f and g is defined as follows.
Example. f(x) x2 2 g(x)
x2 1
71
An example showing reduction in domain

• Let
• The quotient of f and g is
• The domain of f is 0, ?) and the domain of g is
  1, 1.
• The intersection of these domains is 0, 1, but
  we must also exclude x 1 since g(1) 0.
• Therefore, the domain of is 0, 1).

72
Composition of functions
The composition of the function f with the
function g is The domain of
is the set of all x in the domain of g such that
g(x) is in the domain of f.
Example. f(x) x2 1 g(x) x
1
Note In this example, the graph of
is the same as the graph of f shifted left by 1
unit.
73
Composition of Functions--Example

• Suppose we have two money machines, both of which
  increase any money inserted into them. Machine A
  doubles our money while Machine B adds five
  dollars. The money that comes out is described
  by a(x) 2x for Machine A and b(x) x 5 for
  Machine B, where x is the number of dollars
  inserted. The machines can be hooked together so
  that the money coming out of one machine goes
  into the other. There are two ways of hooking up
  the machines which result in the formulas shown
  below. The first formula is while the
  second formula is
• Which of these two compositions would you prefer?
  Why?

A
B


B

A

74
Finding the domain of a composite function--an
example
Domain of f nonnegative real numbers Domain of
g all real numbers Numbers in domain of g such
that g(x) x 1 is nonnegative Domain of
1,?)
75
Decomposition of a given function
If h is a given function, the problem of finding
functions f and g such that h(x) f(g(x)) is
called decomposing. The function f is called the
outer function and the function g is called the
inner function. Example. Find two different
ways of decomposing
1. 2.
76
Inverse functions
Let f and g be two functions such that
f(g(x)) x for every x in the domain of
g, g(f(x)) x for every x in
the domain of f. Under these conditions, the
function g is the inverse function of the
function f. The function g is denoted by f 1
(read "f-inverse"). Therefore,
f(f 1 (x)) x and f 1(f (x))
x. The domain of f must be equal to the range
of f 1, and the range of f must be equal to the
domain of f 1. Note f 1 and are
different. Example. f(x) 2x and f 1(x)
(1/2)x. Note that f 1 "undoes" what f "does".
77
Interchanging input and output Inverse Functions

• The roles of input and output are not necessarily
  fixed. In an earlier example, we derived a
  function f which converts degrees Celsius, C, to
  degrees Fahrenheit, F. The formula for f
  was Suppose now that we
  know the value of F and we wish to compute the
  value of C. We can define a new function g such
  that C g(F). For this function, F is the
  input and C is the output. Of course, g f 1.
  
• Find a formula for the inverse function C f
  1(F). We solve the previous equation for C to
  obtain

78
A function and its inverse "undo" each other.

• What happens if we convert from Celsius to
  Fahrenheit and then back to Celsius? Answer
  We are back where we started. In terms of
  function composition,
• If, on the other hand, we convert from Fahrenheit
  to Celsius and then back to Fahrenheit, we have

79
A function and its inverse "undo" each other,
continued.

• What happens if we convert from Celsius to
  Fahrenheit and then back to Celsius?
• If, on the other hand, we convert from Fahrenheit
  to Celsius and then back to Fahrenheit, we have
• In this example, the domain of f is equal to the
  range of f 1and the range of f is equal to the
  domain of f 1 since they are all real numbers.

80
The graph of an inverse function
The graphs of a function f and its inverse
function are related to each other in the
following way. If the point (a, b) lies on the
graph of f, then the point (b, a) must lie on the
graph of f 1, and vice versa. This means that
the graph of f 1 is a reflection of the graph
of f in the line y x. If we have the graph
of two functions, we can check that one is the
inverse of the other by seeing if the graphs are
the reflections of each other in the line y
x. See the next slide for an example.
81
Verifying inverse functions graphically

• Let f(x) x2 with domain 0, ?). Then f 1(x)
  The squaring function does not have an
  inverse unless its domain is restricted.
• We will plot the graphs of f and f 1 on the same
  coordinate system and show that they are
  reflections of each other in the line y x.

?
?
?
?
82
The Horizontal Line Test for inverse functions

• If there is a horizontal line which intersects a
  functions graph in more than one point, then the
  function does not have an inverse. If every
  horizontal line intersects a functions graph at
  most once, then the function has an inverse. The
  graph of the function f(x) x2, which is shown
  below, fails the horizontal line test. Consider
  y 4.

The line y 4 intersects graph of f(x) x2
twice.
83
One-to-One functions
If every horizontal line intersects a functions
graph at most once, then the function is
one-to-one and the inverse function
exists. Example. f(x) x3 x 1. From the
graph, this function seems to pass the horizontal
line test and is therefore one-to-one. The
inverse function is difficult to find
algebraically but we can use a calculator to
trace along the graph to find individual values
of f 1.
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Finding an inverse function algebraically

• Use the horizontal line test to decide whether f
  has an inverse function.
• In the equation for f(x), replace f(x) by y.
• Interchange the roles of x and y, and solve for
  y.
• Replace y by f 1(x) in the new equation.
• Check your work by showing that the domain of f
  is equal to the range of f 1, the range of f is
  equal to the domain
  of f 1, and f(f 1 (x)) x and f 1(f (x))
  x.

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An example for finding an inverse function

• Consider the rectangle shown (which is defined
  only when x gt 0)
• Let y perimeter of rectangle f(x) 2x 2
  where the domain of f is (0, ?). The graph of
  f is shown next.

1
x
x
1
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Finding the inverse function algebraically

• Consider f(x) 2x 2 with domain (0, ?) and
  range (2, ?).
• 1. f(x) 2x 2 write original
  function
• y 2x 2
• 
  

replace f(x) by y
interchange x and y
solve for y
replace y by f 1(x)
4.
see that inverse formula works
Note Domain of f 1 is (2, ?) and range of f
1 is (0, ?) .
In terms of the rectangle, how do you interpret f
1(100) 49?
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Graphs of f and f 1
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False Inverse function

• If we are given a function y f(x), then the
  function is y g(x) where g(x) ?f(x) is not
  the inverse function of f.
• Example. If is given, then
  is not the inverse of f. The graph of this
  false inverse is obtained by reflecting the graph
  of f in the x-axis.

y x3
y ?x3
89
Inverse functionsa review

• If we are given a function y f(x), then the
  inverse function is the function y g(x)
  satisfying
• f(g(x)) x for every x in the
  domain of g,
• g(f(x)) x for every x in the
  domain of f. As we know,
  the inverse function may fail to exist.
• Example. If is given, then
  is the inverse of f. The graph of the
  inverse is obtained by reflecting the graph of f
  in the line y x.