- Text
- Functional,
- Equations,
- Mathematical,
- Database,
- Equation,
- Variable,
- Continuous,
- Substituting,
- Integers,
- Functions,
- Mathdb.org

1. Functional equations in one variable

Solution. x + 1 Let t = , then x 1 x = t − 1 . Directly substitution yields, () f t 2 ⎛ 1 ⎞ ⎜ ⎟ + 1 ⎝t −1⎠ 1 ⎛ 1 ⎞ 1 ⎜ ⎟ ⎝t −1⎠ t −1 2 = + = − 2 t t+ **1.** Mathematical Database Thus 2 f ( x) = x − x+ **1.** Example **1.**3. 2 If f (ln x) = x + x+ 1, where x > 0 , f**in**d f ( x ). Solution. Let t = ln x, then x t = e . Directly substitution yields, t ( ) 2 t 2x x f() t = e + e + **1.** Thus f( x) = e + e + **1.** In general, if we have f ( g( x)) = h( x) and g( x) has an **in**verse function, then we may replace g 1 ( x) and get f ( x) = h( g ( x)) . − −1 x by Solv**in**g equation Sometimes after perform**in**g transformation of **variable**s, we can arrive at simultaneous **equations**. We can f**in**d the unknown function after solv**in**g the simultaneous **equations**. We may also treat the unknown function as a **variable** **in** ord**in**ary **equations** and solve it. Example **1.**4. If f ( x) 4+ x = 2 3 + f ( x) x 2 , f**in**d f ( x ). Solution. It is equivalent to 2 2 x f x x f x ( ) = (4 + )(3 + ( )) . Simplify**in**g it, we have Page 2 of 10

2 2 2 x f x = + x + + x f x ( ) 3(4 ) (4 ) ( ) − = + 2 4 f( x) 3(4 x ) f( x) =− 2 3(4 + x ) 4 Mathematical Database Example **1.**5. If ⎛ x −1⎞ f ( x) + f ⎜ ⎟= 1+ x, f**in**d f ( x ). ⎝ x ⎠ Solution. Let x −1 t = , then x 1 x = 1 − t . Direct substitution yields f ⎛ 1 ⎞ 1 ⎜ ⎟ + f() t = 1 + ⎝1− t⎠ 1− t (**1.**1) ⎛ 1 ⎞ 1 f ⎜ ⎟ + f( x) = 1 + ⎝1− x ⎠ 1− x Let 1 t = 1 − x , then t −1 x = . Direct substitution yields t (**1.**2) ⎛t−1⎞ ⎛ 1 ⎞ t−1 f ⎜ ⎟+ f ⎜ ⎟= 1+ ⎝ t ⎠ ⎝1− t⎠ t ⎛ x −1⎞ ⎛ 1 ⎞ x −1 f ⎜ ⎟+ f ⎜ ⎟= 1+ ⎝ x ⎠ ⎝1− x⎠ x On the other hand, by substitut**in**g it **in**to (1), we have ⎛ x −1⎞ (**1.**3) f ( x) + f ⎜ ⎟= 1+x ⎝ x ⎠ ((**1.**1) + (**1.**2) + (**1.**3))/2 : ⎛ x−1⎞ ⎛ 1 ⎞ 1⎛ 1 x−1 ⎞ (**1.**4) f ( x) + f ⎜ ⎟+ f ⎜ ⎟= ⎜3+ + + x ⎟⎠ ⎝ x ⎠ ⎝1−x⎠ 2⎝ 1−x x By subtract**in**g (**1.**2) from (**1.**4), we have 3 2 1⎛ 1 x−1 ⎞ ⎛ x−1⎞ 1⎛ 1 x−1⎞ − x + x + 1 f( x) = ⎜3+ + + x⎟− ⎜1+ ⎟= ⎜1+ + x− ⎟= 2⎝ 1−x x ⎠ ⎝ x ⎠ 2⎝ 1−x x ⎠ 2 x(1 −x) Page 3 of 10

- Page 1: Mathematical Database FUNCTIONAL EQ
- Page 5 and 6: Solution. Putting y = 1, then Mathe
- Page 7 and 8: If f ( x ) =−x, from (2.4) we hav
- Page 9 and 10: Mathematical Database Now, as f is