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Inverse Functions and their Representations
Lesson 5.2
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Definition
A function is a set of ordered pairs with no two first elements alike. f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) }
But ... what if we reverse the order of the pairs? This is also a function ... it is the inverse function f -1(x) = { (x,y) : (2, 3), (4, 1), (6, 7), (12, 9) }
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Example
Consider an element of an electrical circuit which increases its resistance as a function of temperature.
T = Temp R = Resistance
-20 50
0 150
20 250
40 350
R = f(T)
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Example
We could also take the view that we wish to determine T, temperature as a function of R, resistance.
R = Resistance T = Temp
50 -20
150 0
250 20
350 40
T = g(R)
Now we would say that g(R) and f(T) are inverse
functions
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Terminology
If R = f(T) ... resistance is a function of temperature,
Then T = f-1(R) ... temperature is the inverse function of resistance. f-1(R) is read "f-inverse of R“ is not an exponent it does not mean reciprocal 1 1
xx
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Does This Have An Inverse? Given the function at the right
Can it have an inverse? Why or Why Not?
NO … when we reverse the ordered pairs, the result is Not a function We would say the function is
not one-to-one
A function is one-to-onewhen different inputs always result in different outputs
x Y
1 5
2 9
4 6
7 5( ) ( )c d f c f d
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One-to-One Functions
When different inputs produce the same output Then an inverse of the function does not exist
When different inputs produce different outputs Then the function is said to be “one-to-one”
Every one-to-one function has an inverse Contrast
( ) ( )c d f c f d
2 3( ) and ( )f x x g x x
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One-to-One Functions
Examples
Horizontal line test?
3( )f x x
2( )f x x
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Finding the Inverse
Try2
2
xy
x
1
Given ( ) 2 7
then 2 7
7solve for x x
27
2
f x x
y x
y
yf y
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Composition of Inverse Functions
Consider f(3) = 27 and f -1(27) = 3 Thus, f(f -1(27)) = 27 and f -1(f(3)) = 3
In general f(f -1(n)) = n and f -1(f(n)) = n
(assuming both f and f -1 are defined for n)
3 1 3( ) ( )f x x and f x x
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Graphs of Inverses
Again, consider Set your calculator for the functions shown
3 1 3( ) ( )f x x and f x x
Dotted style
Use Standard Zoom
Then use Square Zoom
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Graphs of Inverses
Note the two graphs are symmetric about the line y = x
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Investigating Inverse Functions
Consider
Demonstrate that these are inverse functions What happens with f(g(x))? What happens with g(f(x))?
3
3
( ) 2 4
( ) 48
f x x
xg x
Define these functions on your calculator and try
them out
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Domain and Range
The domain of f is the range of f -1 The range of f is the domain of f -1
Thus ... we may be required to restrict the
domain of f so that f -1 is a function
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Domain and Range
Consider the function h(x) = x2 - 9 Determine the inverse function
Problem => f -1(x) is not a function
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Assignment
Lesson 5.2 Page 396 Exercises 1 – 93 EOO