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Transcript of [email protected] MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 1 Bruce Mayer, PE Chabot...
[email protected] • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§9.2b§9.2bInverse FcnsInverse Fcns
[email protected] • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt2
Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §9.2 → Composite Functions
Any QUESTIONS About HomeWork• §9.2 → HW-43
9.2 MTH 55
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Bruce Mayer, PE Chabot College Mathematics
Inverse & One-to-One FunctionsInverse & One-to-One Functions
Let’s view the following two functions as relations, or correspondences:
Maine 1
Illinois 7
Iowa 2
Ohio 3
Domain Range
(inputs) (outputs)
States
ball Ann
rope Jim
phone Jack
car
Domain Range
(inputs) (outputs)
Toys
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Bruce Mayer, PE Chabot College Mathematics
Inverse & One-to-One FunctionsInverse & One-to-One Functions
Suppose we reverse the arrows. We obtain what is called the inverse relation. Are these inverse relations functions?
Maine 1
Illinois 7
Iowa 2
Ohio 3
Range Domain
(inputs) (outputs)
States
ball Ann
rope Jim
phone Jack
Car
Range Domain
(inputs) (outputs)
Toys
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Bruce Mayer, PE Chabot College Mathematics
Inverse & One-to-One FunctionsInverse & One-to-One Functions
Maine 1
Illinois 7
Iowa 2
Ohio 3
Range Domain
(inputs) (outputs)
States
ball Ann
rope Jim
phone Jack
Car
Range Domain
(inputs) (outputs)
Toys
Recall that for each input, a function provides exactly one output. The inverse of “States” correspondence IS a function, but the inverse of “Toys” is NOT.
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Bruce Mayer, PE Chabot College Mathematics
One-to-One for “States” FcnOne-to-One for “States” Fcn
In the States function, different inputs have different outputs, so it is a one-to-one function.
In the Toys function, rope and phone are both paired with Jim.
Thus the Toy function is NOT one-to-one.
ball Ann
rope Jim
phone Jack
Car
Range Domain
(inputs) (outputs)
Toys
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Bruce Mayer, PE Chabot College Mathematics
One-to-One SummarizedOne-to-One Summarized
A function f is one-to-one if different inputs have different outputs. That is, if for a and b in the domain of f with a ≠ b we have f(a) ≠ f(b) then the function f is one-to-one.
If a function is one-to-one, then its INVERSE correspondence is ALSO a FUNCTION.
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Bruce Mayer, PE Chabot College Mathematics
One-to-One Fcn GraphicallyOne-to-One Fcn Graphically
Each y-value in the range corresponds to only one x-value in the domain
• i.e.; Each x has a Unique y
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Bruce Mayer, PE Chabot College Mathematics
NOT a One-to-One FcnNOT a One-to-One Fcn
The y-value y2 in the range corresponds to TWO x-values, x2 and x3, in the domain.
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Bruce Mayer, PE Chabot College Mathematics
NOT a Function at AllNOT a Function at All
The x-value x2 in the domain corresponds to the TWO y-values, y2 and y3, in the range.
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Bruce Mayer, PE Chabot College Mathematics
Definition of Inverse FunctionDefinition of Inverse Function
Let f represent a one-to-one function. The inverse of f is also a function, called the inverse function of f, and is denoted by f−1.
If (x, y) is an ordered pair of f, then (y, x) is an ordered pair of f−1, and we write x = f−1(y). We have y = f (x) if and only if f−1(y) = x.
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Bruce Mayer, PE Chabot College Mathematics
Example Example ff-values ↔ -values ↔ ff-1-1-values-values
Assume that f is a one-to-one function.a. If f(3) = 5, find f-1(5)
b. If f-1(−1) = 7, find f(7)
Solution: Recall that y = f(x) if and only if f-1(y) = x
a. Let x = 3 and y = 5. Now 5 = f(3) if and only if f−1(5) = 3. Thus, f−1(5) = 3.
b. Let y = −1 and x = 7. Now, f−1(−1) = 7 if and only if f(7) = −1. Thus, f (7) = −1.
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Bruce Mayer, PE Chabot College Mathematics
Inverse Function PropertyInverse Function Property
Let f denote a one-to-one function. Then
f o f 1 x f f 1 x x
for every x in the domain of f–1.
1.
f 1 o f x f 1 f x x
for every x in the domain of f .
2.
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Bruce Mayer, PE Chabot College Mathematics
Example Example Inverse Fcn Property Inverse Fcn Property
Let f(x) = x3 + 1. Show that 1 3( ) 1.f x x
Soln: 1 1( ) ( )f f x f f x
1 1( ) ( )f f x f f x
3 1f x
33 1 1x
1 1x x
1 3 1f x
33 ( 1) 1x
3 3x x
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Bruce Mayer, PE Chabot College Mathematics
UNIQUE Inverse Fcn PropertyUNIQUE Inverse Fcn Property
Let f denote a one-to-one function. Then if g is any function such that
g = f –1. That is, g is the inverse function of f.
f g x x for every x in the domain of g and
g f x x for every x in the domain of f, then
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Bruce Mayer, PE Chabot College Mathematics
Verify Inverse FunctionsVerify Inverse Functions
Verify that the following pairs of functions are inverses of each other:
f x 2x 3 and g x x 3
2.
Solution: From the composition of f & g.
f og x f g x fx 3
2
2x 3
2
3 x 3 3
x
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Bruce Mayer, PE Chabot College Mathematics
Verify Inverse FunctionsVerify Inverse Functions
Solution (cont.): Now Find
Observe:
g f x .
g o f x g f x g 2x 3
2x 3 3
2x
f g x g f x x,
This Verifies that f and g are indeed inverses of each other.
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Bruce Mayer, PE Chabot College Mathematics
Example Example Find Inverse of a Fcn Find Inverse of a Fcn
Given that f(x) = 5x − 2 is one-to-one, then find an equation for its inverse
Solution: f (x) = 5x – 2
y = 5x – 2
x = 5y – 2
2
5
xy
1 2( )
5
xf x
Replace f(x) with y
Interchange x and y
Solve for y
Replace y with f-1(x)
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Bruce Mayer, PE Chabot College Mathematics
Procedure for finding Procedure for finding ff−−11
1. Replace f(x) by y in the equation for f(x).
2. Interchange x and y.
3. Solve the equation in Step 2 for y.
4. Replace y with f−1(x).
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Bruce Mayer, PE Chabot College Mathematics
Example Example Find the Inverse Find the Inverse
Find the inverse of the one-to-one function
Solution: y x 1
x 2Step 1
x y 1
y 2Step 2
Step 3 12
12
yxxy
yyx
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Bruce Mayer, PE Chabot College Mathematics
Example Example Find the Inverse Find the Inverse
Step 3(cont.)
f 1 x 2x 1
x 1, x 1Step 4
yxyyxxxy 2122
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Bruce Mayer, PE Chabot College Mathematics
Example Example Find Domain & Range Find Domain & Range
Find the Domain &Range of the function
Solution: Domain of f, all real numbers x such that x ≠ 2, in interval notation (−∞, 2)U(2, −∞)
Range of f is the domain of f−1 f 1 x 2x 1
x 1, x 1
Range of f is (−∞, 1) U (1, −∞)
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Bruce Mayer, PE Chabot College Mathematics
Inverse Function MachineInverse Function Machine
Let’s consider inverses of functions in terms of function machines. Suppose that a one-to-one function f, has been programmed into a machine.
If the machine has a reverse switch, when the switch is thrown, the machine performs the inverse function, f−1. Inputs then enter at the opposite end, and the entire process is reversed.
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Bruce Mayer, PE Chabot College Mathematics
Reverse Switch GraphicallyReverse Switch Graphically
Forward
Reverse
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Bruce Mayer, PE Chabot College Mathematics
Horizontal Line TestHorizontal Line Test Recall that to be a
Function an (x,y) relation must pass the VERTICAL LINE test
In order for a function to have an inverse that is a function, it must pass the HORIZONTAL-LINE test as well
NOT a Function – Fails the Vertical Line Test
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Bruce Mayer, PE Chabot College Mathematics
Horizontal Line Test DefinedHorizontal Line Test Defined If it is impossible to
draw a horizontal line that intersects a function’s graph more than once, then the function isone-to-one.
For every one-to-one function, an inverse function exists.
A Function withOUT and Inverse – Fails the Horizontal Line Test (not 1-to-1)
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Bruce Mayer, PE Chabot College Mathematics
Example Example Horizontal Line Test Horizontal Line Test Determine whether the function f(x) = x2 + 1 is
one-to-one and thus has an inverse fcn. The graph of f is shown. Many
horizontal lines cross the graph more than once. For example, the line y = 2 crosses where the first coordinates are 1 and −1. Although they have different inputs, they have the same output: f(−1) = 2 = f(1). The function is NOT one-to-one, therefore NO inverse function exists
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
4
3
6
2
5
1
-1
-2
78
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Bruce Mayer, PE Chabot College Mathematics
Example Example Horizontal Ln Test Horizontal Ln Test
Use the horizontal-line test to determine which of the following fcns are 1-to-1
a. b.
Soln a. • No horizontal line
intersects the graph of f in more than one point, therefore the function f is one-to-one
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Bruce Mayer, PE Chabot College Mathematics
Example Example Horizontal Ln Test Horizontal Ln Test
Use the horizontal-line test to determine which of the following fcns are 1-to-1
a. b.
Soln b. • No horizontal line
intersects the graph of f in more than one point, therefore the function f is 1-to-1
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Bruce Mayer, PE Chabot College Mathematics
Graphing Fcns and Their InversesGraphing Fcns and Their Inverses
How do the graphs of a function and its inverse compare?
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Bruce Mayer, PE Chabot College Mathematics
Example Example Graphs Inverse Graphs Inverse FcnFcn Graph f(x) = 5x − 2 and f−1(x) = (x + 2)/5
on the same set of axes and compare Solution:
x -5 -4 -3 -2 -1 1 2 3 4 5
-3
2
-2
3
-1
1
6
54
-4
-5
f (x) = 5x – 2
f -1(x) = (x + 2)/5
Note that the graph of f−1(x) can be drawn by reflecting the graph of f across the line y = x.
When x and y are interchanged to find a formula for f−1(x), we are, in effect, Reflecting or Flipping the graph of f.
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Bruce Mayer, PE Chabot College Mathematics
Visualizing Inverses Visualizing Inverses
The graph of f−1 is a REFLECTION of the graph of f across the line y = x.
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Bruce Mayer, PE Chabot College Mathematics
Example Example Use Use yy = = xx Mirror Ln Mirror Ln
The graph of the function f is shown at Lower Right. Sketch the graph of the f−1
Soln
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Bruce Mayer, PE Chabot College Mathematics
Example Example Inverse or Not? Inverse or Not? Ray’s Music Mart has six employees. The
first table lists the first names and the Social Security numbers of the employees, and the second table lists the first names and the ages of the employees
a. Find the inverse of the function defined by the first table, and determine whether the inverse relation is a function
b. Find the inverse of the function defined by the second table, and determine whether the inverse relation is a function
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Bruce Mayer, PE Chabot College Mathematics
Example Example Inverse or Not? Inverse or Not?
Dwayne 590-56-4932
Sophia 599-23-1746
Desmonde 264-31-4958
Carl 432-77-6602
Anna 195-37-4165
Sal 543-71-8026
Solution:Every y-value corresponds to exactly one x-value. Thus the inverse of the function defined in this table is a function
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Bruce Mayer, PE Chabot College Mathematics
Example Example Inverse or Not? Inverse or Not?
Solution:There is more than one x-value that corresponds to a y-value. For example, the age of 24 yields the names Dwayne and Anna. Thus the inverse of the function defined in this table is NOT a function.
Dwayne 24
Sophia 26
Desmonde 42
Carl 51
Anna 24
Sal 26
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Bruce Mayer, PE Chabot College Mathematics
Example Example Hydrostatic Pressure Hydrostatic Pressure
The formula for finding the water pressure p (in pounds per square inch, or psi),at a depth d (in feet) below the surface is
A pressure gauge on a Diving Bell breaks and shows a reading of 1800 psi. Determine how far below the surface the bell was when the gauge failed
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Bruce Mayer, PE Chabot College Mathematics
Example Example HydroStatic P HydroStatic P
Solution: The depth is given by the inverse of
Solve theInverseEqn for p
p 15d
3333p 15d
d 33p
15
Let p = 1800 psi
d 33 1800
15d 3960
The Diving Bell was 3960 feet below the surface when the gauge failed
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Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §9.2 Exercise Set• 38, 42, 60, 68, 76
Some Temperature Scales
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Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
Old StyleDiving
Bell
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22