Difference between revisions of "Functional equation"
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*<math>f(x) + 2f\left(\frac1x\right) = 2x</math> | *<math>f(x) + 2f\left(\frac1x\right) = 2x</math> | ||
*<math>g(x)^2 + 4g(x) + 4 = 8\sin{x}</math> | *<math>g(x)^2 + 4g(x) + 4 = 8\sin{x}</math> | ||
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==Introductory Topics== | ==Introductory Topics== | ||
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The inverse of a function is a function that "undoes" a function. For an example, consider the function: f(x)<math> = 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>. | The inverse of a function is a function that "undoes" a function. For an example, consider the function: f(x)<math> = 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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So, clearly, <math>f(y) = \frac{3}{25}y^2 - \frac{4}{25y^2}</math> | So, clearly, <math>f(y) = \frac{3}{25}y^2 - \frac{4}{25y^2}</math> | ||
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| + | === Problem Examples === | ||
| + | * [[2006_AMC_12A_Problems/Problem_18 | 2006 AMC 12A Problem 18]] | ||
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==See Also== | ==See Also== | ||
*[[Functions]] | *[[Functions]] | ||
*[[Polynomials]] | *[[Polynomials]] | ||
Revision as of 02:11, 12 July 2006
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: f(x)
. 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 subtract the second equation from the first, we have:
So, clearly, 
Problem Examples
