From fb1cc3837c3ec39dbb954b562fda5ba00ba34e3b Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Fatih=20Ayd=C4=B1n?= <unifaydin@protonmail.com>
Date: Mon, 28 Sep 2020 20:06:03 +0000
Subject: [PATCH] Small corrections in 'The Dual of a function'

---
 ...20-09-28-an-indepth-look-at-duals-and-their-circuits.html.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/source/articles/2020-09-28-an-indepth-look-at-duals-and-their-circuits.html.md b/source/articles/2020-09-28-an-indepth-look-at-duals-and-their-circuits.html.md
index b9ecf01..38b7d17 100644
--- a/source/articles/2020-09-28-an-indepth-look-at-duals-and-their-circuits.html.md
+++ b/source/articles/2020-09-28-an-indepth-look-at-duals-and-their-circuits.html.md
@@ -103,7 +103,7 @@ Similarly here are some examples of transformations and their inverse that are t
 
 # The Dual of a function
 
-Just as we have shown above that individual variables and values have a dual under an invertible function, likewise functions can also have duals in the same manner. Imagine we have an invertible function \\(T(x)\\) which will convert something to its dual, and we have some function \\(f(x)\\) we wish to find the dual of, then simply by passing the function into T we can produce its dual. Specifically \\(T(f(x)) = f^T(x)\\) where the functions \\(f(x)\\) and \\(f^T(x)\\) are duals of each other. It is important to note here only \\(T(x)\\) need be invertible, neither \\(f(x)\\) nor \\(f^T(x)\\) need to have this property. For example, say the transformation under which we create the duals is the reciprocal function, which is invertible, but \\(f(x)\\) is the square function, which is not invertible. We know it isn't invertable because 10 squared is 100 and -10 squared is also 100. So there is no way to reverse the value of 100 and get the original value since some information was lost; we no longer know if the original value was positive or negative.
+Just as we have shown above that individual variables and values have a dual under an invertible function, likewise functions can also have duals in the same manner. Imagine we have an invertible function \\(T(x)\\) which will convert something to its dual, and we have some function \\(f(x)\\) we wish to find the dual of, then simply by passing the function into T we can produce its dual. Specifically \\(T(f(x)) = f^T(x)\\) where the functions \\(f(x)\\) and \\(f^T(x)\\) are duals of each other. It is important to note here only \\(T(x)\\) needs be invertible, neither \\(f(x)\\) nor \\(f^T(x)\\) needs to have this property. For example, say the transformation under which we create the duals is the reciprocal function, which is invertible, but \\(f(x)\\) is the square function, which is not invertible. We know it isn't invertible because 10 squared is 100 and -10 squared is also 100. So there is no way to reverse the value of 100 and get the original value since some information was lost, we no longer know if the original value was positive or negative.
 
 $$T(x) = \frac{1}{X}$$
 
-- 
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