Difference between revisions of "Functional equation"
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===The Inverse of a Function=== | ===The Inverse of a Function=== | ||
− | The inverse of a function is a function that "undoes" a function. For an example, consider the function: f(x) | + | The inverse of a function is a function that "undoes" a function. For an example, consider the function: <math>f(x) = x^2 + 6</math>. The function <math>g(x) = \sqrt{x-6}</math> has the property that <math>f(g(x)) = x</math>. In this case, <math>g</math> is called the '''(right) inverse function'''. (Similarly, a function <math>g</math> so that <math>g(f(x))=x</math> is called the '''left inverse function'''. Typically the right and left inverses coincide on a suitable domain, and in this case we simply call the right and left inverse function the '''inverse function'''.) Often the inverse of a function <math>f</math> is denoted by <math>f^{-1}</math>. |
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==Intermediate Topics== | ==Intermediate Topics== | ||
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<math>3f\left(\frac1y\right)- 4f(y) = \frac1{y^2}</math> | <math>3f\left(\frac1y\right)- 4f(y) = \frac1{y^2}</math> | ||
− | Now, if we multiply the first equation by 3 and the second equation by 4, and | + | Now, if we multiply the first equation by 3 and the second equation by 4, and add the two equations, we have: |
− | <math> | + | <math>-7f(y) = 3y^2 + \frac{4}{y^2}</math> |
− | So, clearly, <math>f(y) = \frac{3}{ | + | So, clearly, <math>f(y) = -\frac{3}{7}y^2 - \frac{4}{7y^2}</math> |
=== Problem Examples === | === Problem Examples === | ||
* [[2006_AMC_12A_Problems/Problem_18 | 2006 AMC 12A Problem 18]] | * [[2006_AMC_12A_Problems/Problem_18 | 2006 AMC 12A Problem 18]] | ||
* [[2007_AIME_II_Problems/Problem_14 | 2007 AIME II Problem 14]] | * [[2007_AIME_II_Problems/Problem_14 | 2007 AIME II Problem 14]] | ||
+ | |||
+ | ==Advanced Topics== | ||
+ | |||
+ | ===Injectivity and Surjectivity=== | ||
==See Also== | ==See Also== | ||
*[[Functions]] | *[[Functions]] | ||
− | *[[ | + | *[[Cauchy Functional Equation]] |
− | [[Category: | + | [[Category:Algebra]] |
− | [[Category: | + | [[Category:Definition]]] |
Revision as of 14:40, 24 August 2021
A functional equation, roughly speaking, is an equation in which some of the unknowns to be solved for are functions. For example, the following are functional equations:
Contents
Introductory Topics
The Inverse of a Function
The inverse of a function is a function that "undoes" a function. For an example, consider the function: . The function has the property that . In this case, is called the (right) inverse function. (Similarly, a function so that is called the left inverse function. Typically the right and left inverses coincide on a suitable domain, and in this case we simply call the right and left inverse function the inverse function.) Often the inverse of a function is denoted by .
Intermediate Topics
Cyclic Functions
A cyclic function is a function that has the property that:
A classic example of such a function is because . Cyclic functions can significantly help in solving functional identities. Consider this problem:
Find such that . In this functional equation, let and let . This yields two new equations:
Now, if we multiply the first equation by 3 and the second equation by 4, and add the two equations, we have:
So, clearly,