1. Functional equations in one variable
1. Functional equations in one variable
1. Functional equations in one variable
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Solution.<br />
Putt<strong>in</strong>g<br />
y = 1, then<br />
Mathematical Database<br />
f( x) = f( x) f(1) − f( x+ 1) + 1<br />
= 2 f( x) − f( x+ 1) + 1<br />
f( x+ 1) = f( x) + 1<br />
Therefore apply<strong>in</strong>g condition 1 and by mathematical <strong>in</strong>duction, for all <strong>in</strong>teger x, we have<br />
f ( x) = x+ <strong>1.</strong><br />
m<br />
m<br />
For any rational number, let x = where m, n are <strong>in</strong>tegers and n is not zero. Putt<strong>in</strong>g x = , y = n,<br />
n<br />
n<br />
⎛<br />
then ( ) m ⎞ ⎛m<br />
⎞<br />
f m = f ⎜ ⎟( n+ 1) − f ⎜ + n⎟<br />
1<br />
⎝ n ⎠ ⎝ n ⎠ + .<br />
S<strong>in</strong>ce f( x+ 1) = f( x)<br />
+ 1 for ∀x<br />
∈ , we have<br />
⎛m<br />
⎞ ⎛m⎞<br />
f ⎜ + n⎟= f ⎜ ⎟+n.<br />
⎝ n ⎠ ⎝ n ⎠<br />
⎛m⎞ ⎛m⎞<br />
Substitut<strong>in</strong>g this <strong>in</strong>to the orig<strong>in</strong>al equation, we have m+ 1 = f ⎜ ⎟( n+ 1) − f ⎜ ⎟−n+ <strong>1.</strong> Thus<br />
⎝ n ⎠ ⎝ n ⎠<br />
⎛m⎞ m<br />
f ⎜ ⎟ = + <strong>1.</strong><br />
⎝ n ⎠ n<br />
So we have f ( x) = x+ 1 ∀x<br />
∈ .<br />
Remark.<br />
In this question we can see that we first solve the functional equation for a special case (we found<br />
f ( x ) when x ∈ ), then we solve for the more general case. (we found f ( x ) when x ∈ ). Indeed,<br />
this method is used <strong>in</strong> many occasions. For<br />
f<strong>in</strong>d<br />
when<br />
f ( x ) when<br />
x∈<br />
about this later.<br />
x ∈ . Then, f<strong>in</strong>d f ( x ) when x ∈ by substitut<strong>in</strong>g<br />
f : →<br />
, we can apply this method similarly. First,<br />
m<br />
x = . F<strong>in</strong>ally, f<strong>in</strong>d f ( x )<br />
n<br />
by the density of rational numbers (for cont<strong>in</strong>uous functions only). We shall see more<br />
Example 2.2.<br />
If<br />
2 2<br />
( x y) f( x y) ( x y) f( x y) 4 xy( x y )<br />
− + − + − = − for all x,<br />
y , f<strong>in</strong>d f ( x ).<br />
Solution.<br />
The given condition is equivalent to<br />
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