Mae M
composite functions
Unit 2 Discussion
Start!
what are composite funcitons?
Composite functions, like the name, are functions composing of other functions. These functions often take the form f[g(x)], read as "f of g of x". Since compostie functions invlove one function inside another, the process of multiplication can be used to simplify them. For example: f [ g (x) ] = ( f * g ) ( x )
Composite Function Example
Below is the simplified version of f [ g ( x ) ] when f (x) = 2x - 10 and g (x) = -4x -2 .
1. f [ g (x) ] = ( f * g ) (x)2. f [ g (x) ] = ( (2x - 10) * (-4x-2) ) 3. f [ g (x) ] = -8x + 4x + 40x + 20 4. f [ g (x) ] = 44x - 8x + 20 5. f [ g (x) ] = 36x + 20
KEY:green - composite function red - f ( x ) blue - g ( x )
1. Expand the composite function into its two defined parts2. Substitute f and g for the given functions 3. Multiply using FOIL (first, outer, inner, last) 4. Combine like terms 5. The end!
SInce all of these lines can be passed through by a vertical line with only one intersection, this means that both the original equations and their composite are functions!
Inverses of functions
Let's start with the same two functions from before, f ( x ) = 2x - 10 and g ( x ) = -4x -2, and find their inverse functions. Inverse functions - by definition - have the same ordered pairs as a function, but the coordinates are interchanged. An inverse function is modeled by f^-1 ( x ); read as "the inverse of f of x".
1. change g ( x ) to y 2. interchange x and y 3. isolate y 4. simplify
f ( x ) = 2x - 101. y = 2x - 10 2. x = 2y - 10 3. -2y = -x - 10 4. y = 0.5x + 5 5. f^-1 ( x ) = 0.5x + 5
g ( x ) = -4x - 21. y = -4x - 2 2. x = -4y - 2 3. 4y = -x - 2 4. y = -0.25x - 0.5 5. g^-1 ( x ) = -0.25x - 0.5 5. replace y with g^-1 ( x )
1. change f ( x ) to y 2. interchange x and y 3. isolate y 4. simplify 5. replace y with f^-1 ( x )
REMEMBER!The -1 in f^-1 ( x ) does NOT indicate an exponent, it simply just shows that it is an inverse.
g^-1 ( x ) = 0.25x - 0.5
Graphs of Inverses
f^-1 ( x ) = 0.5x + 5
IS A FUNCTION!
IS A FUNCTION!
To find the answer to this question, we can utilize the composite function f [g ( x )] from earlier: f [ g (x) ] = 36x + 20 when f ( x ) = 2x - 10 and g ( x ) = -4x - 2. Just like before with f^-1 ( x ) and g^-1 ( x ), we can use the same steps to find the inverse of the composite function f^-1 [ g (x) ]. 1. f [ g (x) ] = 36x + 20 1. original composite function 2. y = 36x + 20 2. replace f[g(x)] with y 3. x = 36y + 20 3. interchange x and y 4. -36y = -x + 20 4. isolate y 5. y = 1/36x - 5/9 5. simplify 6. f^-1 [ g (x) = 1/36x - 5/9 6. replace y with f^-1[g(x)] So, the inverse composite function of f ( x ) and g ( x ) is f^-1 [ g (x) = 1/36x = 5/9
Conclusion (part I)
If the inverses of two functions are both functions, will the inverse of the composite function made by the original functions also be a function?
Conclusion (part II)
f [ g (x) ] and f^-1 [ g (x) ]
f ( x ) and f^-1 ( x )
are functions!
both the composite and the inverse of the composite are functions; they both pass the vertical line test!
g ( x ) and g^-1 ( x )
are functions!
If the inverses of two functions are both functions, will the inverse of the composite function made by the original functions also be a function?
YES! With the example of the inverse, composite, and composite inverse of f ( x ) and g ( x ), it is proven that the inverse of the composite function made by the original functions is a function.
Thanks!
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Transcript
Mae M
composite functions
Unit 2 Discussion
Start!
what are composite funcitons?
Composite functions, like the name, are functions composing of other functions. These functions often take the form f[g(x)], read as "f of g of x". Since compostie functions invlove one function inside another, the process of multiplication can be used to simplify them. For example: f [ g (x) ] = ( f * g ) ( x )
Composite Function Example
Below is the simplified version of f [ g ( x ) ] when f (x) = 2x - 10 and g (x) = -4x -2 .
1. f [ g (x) ] = ( f * g ) (x)2. f [ g (x) ] = ( (2x - 10) * (-4x-2) ) 3. f [ g (x) ] = -8x + 4x + 40x + 20 4. f [ g (x) ] = 44x - 8x + 20 5. f [ g (x) ] = 36x + 20
KEY:green - composite function red - f ( x ) blue - g ( x )
1. Expand the composite function into its two defined parts2. Substitute f and g for the given functions 3. Multiply using FOIL (first, outer, inner, last) 4. Combine like terms 5. The end!
SInce all of these lines can be passed through by a vertical line with only one intersection, this means that both the original equations and their composite are functions!
Inverses of functions
Let's start with the same two functions from before, f ( x ) = 2x - 10 and g ( x ) = -4x -2, and find their inverse functions. Inverse functions - by definition - have the same ordered pairs as a function, but the coordinates are interchanged. An inverse function is modeled by f^-1 ( x ); read as "the inverse of f of x".
1. change g ( x ) to y 2. interchange x and y 3. isolate y 4. simplify
f ( x ) = 2x - 101. y = 2x - 10 2. x = 2y - 10 3. -2y = -x - 10 4. y = 0.5x + 5 5. f^-1 ( x ) = 0.5x + 5
g ( x ) = -4x - 21. y = -4x - 2 2. x = -4y - 2 3. 4y = -x - 2 4. y = -0.25x - 0.5 5. g^-1 ( x ) = -0.25x - 0.5 5. replace y with g^-1 ( x )
1. change f ( x ) to y 2. interchange x and y 3. isolate y 4. simplify 5. replace y with f^-1 ( x )
REMEMBER!The -1 in f^-1 ( x ) does NOT indicate an exponent, it simply just shows that it is an inverse.
g^-1 ( x ) = 0.25x - 0.5
Graphs of Inverses
f^-1 ( x ) = 0.5x + 5
IS A FUNCTION!
IS A FUNCTION!
To find the answer to this question, we can utilize the composite function f [g ( x )] from earlier: f [ g (x) ] = 36x + 20 when f ( x ) = 2x - 10 and g ( x ) = -4x - 2. Just like before with f^-1 ( x ) and g^-1 ( x ), we can use the same steps to find the inverse of the composite function f^-1 [ g (x) ]. 1. f [ g (x) ] = 36x + 20 1. original composite function 2. y = 36x + 20 2. replace f[g(x)] with y 3. x = 36y + 20 3. interchange x and y 4. -36y = -x + 20 4. isolate y 5. y = 1/36x - 5/9 5. simplify 6. f^-1 [ g (x) = 1/36x - 5/9 6. replace y with f^-1[g(x)] So, the inverse composite function of f ( x ) and g ( x ) is f^-1 [ g (x) = 1/36x = 5/9
Conclusion (part I)
If the inverses of two functions are both functions, will the inverse of the composite function made by the original functions also be a function?
Conclusion (part II)
f [ g (x) ] and f^-1 [ g (x) ]
f ( x ) and f^-1 ( x )
are functions!
both the composite and the inverse of the composite are functions; they both pass the vertical line test!
g ( x ) and g^-1 ( x )
are functions!
If the inverses of two functions are both functions, will the inverse of the composite function made by the original functions also be a function?
YES! With the example of the inverse, composite, and composite inverse of f ( x ) and g ( x ), it is proven that the inverse of the composite function made by the original functions is a function.
Thanks!