Chapter 4.1 Inverse Functions. Inverse Operations Addition and subtraction are inverse operations...
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Transcript of Chapter 4.1 Inverse Functions. Inverse Operations Addition and subtraction are inverse operations...
![Page 1: Chapter 4.1 Inverse Functions. Inverse Operations Addition and subtraction are inverse operations starting with a number x, adding 5, and subtracting.](https://reader035.fdocuments.in/reader035/viewer/2022062407/56649e0f5503460f94afa18a/html5/thumbnails/1.jpg)
Chapter 4.1
Inverse Functions
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Inverse Operations
Addition and subtraction are inverse operations starting with a number x, adding 5, and subtracting 5 gives x back as the result.
![Page 3: Chapter 4.1 Inverse Functions. Inverse Operations Addition and subtraction are inverse operations starting with a number x, adding 5, and subtracting.](https://reader035.fdocuments.in/reader035/viewer/2022062407/56649e0f5503460f94afa18a/html5/thumbnails/3.jpg)
Similarly some functions are inverses of each other. For example the functions defined by
are inverses of each other with respect to function composition.
xxf 8)( xxg8
1)(
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This means that if the value of x such as x = 12 is chosen, then
96128)12( f
12968
1)96( g
Thus, 1212 fg
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Also,
For these functions f and g, it can be shown that
1212 gf
xxgf xxfg and
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One-to-One Functions
Suppose we define the function
F = { (-2, 2), (-1, 1), (0, 0), (1, 3), (2, 5) }.
We can form another set of ordered pairs, from F by interchanging the x- and y-values of each pair in F. We call this set G, so
G = { (2, -2), (1, -1), (0, 0), (3, 1), (5, 2) }.
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One-to-One Functions
F = { (-2, 2), (-1, 1), (0, 0), (1, 3), (2, 5) }.
G = { (2, -2), (1, -1), (0, 0), (3, 1), (5, 2) }.
To show that these two sets are related, G is called the inverse of F. For a function f to have an inverse, f must be a one-to-one function.
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225)( xxf
Determine whether the function defined by
is one-to-one.
2 25)3( f
2 25)3( f
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Determine whether each graph is the graph of a one-to-one function
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Determine whether each graph is the graph of a one-to-one function
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Notice that the function graphed in the example decreases on its entire domain.
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In general a function that is either increasing or decreasing on its entire domain, such as
must be one-to-one.
,)( xxf ,)( 3xxf
,)( and xxf
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Inverse Functions
As mentioned earlier, certain pairs of one-to-one functions “undo” one another.
For example if
58)( xxf8
5)( and
x
xg
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Inverse Functions
58)( xxf8
5)( and
x
xg
855108)10(then f
108
585)85( and
g
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Inverse Functions
58)( xxf8
5)( and
x
xg
)3(then f
)29( and g
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Inverse Functions
58)( xxf8
5)( and
x
xg
)5(then f
)35( and g
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Inverse Functions
58)( xxf8
5)( and
x
xg
)2(then g
)8
3( and f
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Let functions f and g be defined by
respectively. Is g the inverse function of f?
1)( 3 xxf 3 1)( and xxg
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1)( 3 xxf 3 1)( and xxg
xgf
xfg
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A special notation is often used for inverse functions: If g is the inverse of a function f, then g is written as f-1 (read “f-inverse”). in Example 3.
31 1)( xxf
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Do not confuse the -1 in f-1 with a negative exponent. The symbol f-1 (x) does not represent
It represents the inverse function of f.
;)(
1
xf
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By the definition of inverse function, the domain of f equals the range of f -1,
and the range of f equals the domain of f -1.
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Find the inverse of each function that is one-to-one.
G = { (3, 1), (0, 2), (2, 3), (4, 0) }.
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Find the inverse of each function that is one-to-one.If the Pollutant Standard Index (PSI), an indicator of air quality, exceeds 100 on a particular day, then that day is classified as unhealthy. The table in the margin shows the number of unhealthy days in Chicago for selected years. Let f be the function in the table with the years forming the domain and the numbers of unhealthy days forming the range.
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Find the inverse of each function that is one-to-one.
F = { (-2, 1), (-1, 0), (0, 1), (1, 2), (2, 2) }.
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Find the inverse of each function that is one-to-one.
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Equations of Inverses
By definition, the inverse of a one-to-one function si found by interchanging the x- and y-values of each of its ordered pairs. The equation of the inverse of a function defined by y = f(x) is found in the same way.
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Decide whether each equation defines a one-to-one function. If so, find the equation of the inverse.
f(x) = 2x + 5
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Decide whether each equation defines a one-to-one function. If so, find the equation of the inverse.
f(x) = x2 + 2
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Decide whether each equation defines a one-to-one function. If so, find the equation of the inverse.
f(x) = (x – 2)3
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One way to graph the inverse of a function f whose equation is known is to find some ordered pairs that are on the graph of f, interchange x and y to get ordered pairs that are on the graph of f -1, plot those points, and sketch the graph of f -1 through the points.
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For example, suppose the point (a, b) shown in Figure 7 is on the graph of a one-to-one function f. Then the point (b, a) is on the graph of f -1 .
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The line segment connecting (a, b) and (b, a) is perpendicular to, and cut in half by, the line y = x.
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The points (a, b) and (b, a) are “mirror images” of each other with respect to y = x.
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For this reason we can find the graph of f -1 from the graph of f by locating the mirror image of each point if f with respect to the line y = x.
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Graph the inverses of functions f (shown in blue) in Figure 8.
![Page 42: Chapter 4.1 Inverse Functions. Inverse Operations Addition and subtraction are inverse operations starting with a number x, adding 5, and subtracting.](https://reader035.fdocuments.in/reader035/viewer/2022062407/56649e0f5503460f94afa18a/html5/thumbnails/42.jpg)
Graph the inverses of functions f (shown in blue) in Figure 8.
![Page 43: Chapter 4.1 Inverse Functions. Inverse Operations Addition and subtraction are inverse operations starting with a number x, adding 5, and subtracting.](https://reader035.fdocuments.in/reader035/viewer/2022062407/56649e0f5503460f94afa18a/html5/thumbnails/43.jpg)
Graph the inverses of functions f (shown in blue) in Figure 8.
![Page 44: Chapter 4.1 Inverse Functions. Inverse Operations Addition and subtraction are inverse operations starting with a number x, adding 5, and subtracting.](https://reader035.fdocuments.in/reader035/viewer/2022062407/56649e0f5503460f94afa18a/html5/thumbnails/44.jpg)
Graph the inverses of functions f (shown in blue) in Figure 8.
![Page 45: Chapter 4.1 Inverse Functions. Inverse Operations Addition and subtraction are inverse operations starting with a number x, adding 5, and subtracting.](https://reader035.fdocuments.in/reader035/viewer/2022062407/56649e0f5503460f94afa18a/html5/thumbnails/45.jpg)
x-1f Find ,5x f(x)Let
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An Application of Inverse Functions to Cryptography
A one-to-one function and its inverse can be used to make information secure.
The function si used to encode a message, and its inverse is used to decode the coded message.
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An Application of Inverse Functions to Cryptography
In practice, complicated functions are used. We illustrate the process with the simple function defined by f(x) = 2x + 5.
Each letter of the alphabet is asigned a numerical value according to its position in the alphabet, as follows.
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Use the one-to-one function defined by f(x) = 2x + 5 and find the numerical values repeated in the margin to encode the word ALGEBRA. 5 2 f(1)A
5 2 f(12)L 5 2 f(7)G 5 2 f(5)E 5 2 f(2)B 5 2 f(18)R