Solve函數計算的結果為什麼不能分步轉換成函數？

02-02

使用Solve求解方程，求得的解可以保存成函數使用，但我試了兩種方法，只有第一種起作用，從語義上說完全一樣，只是第二種是分兩步做的，為什麼會這樣？
第一種：

第二種：

方法一中由於 SetDelayed （:=）有 HoldAll 這個屬性，會阻止右手邊的定義表達式中 sol[[1]] 被進一步取值為 y -&> 3-x。你可以把方法一修改為 f[x_] = y /. sol[[1]] 來解決這個問題。Set (=) 和 SetDelayed (:=) 的區別可以看看文檔「立即定義和延時定義」。可以根據適用性選擇用 SetDelayed（:=）定義函數 f[x_] := ... 或用 Set（=）定義函數 f[x] := ... 。

遇到這類貌似迷惑的問題，經常可以用 DownValues 和 TracePrint 來「解謎」。比如你可以用 DownValues[f] 和 TracePrint[f[5]] 觀察一下兩種方法下 f 函數的定義究竟如何不同，以及 f[5] 的求值步驟又具體是怎樣的（若不熟悉這兩個函數，可以看他們的文檔和示例）：

方法一：

In[98]:= sol = Solve[x + y == 3, y]


Out[98]= {{y -&> 3 - x}}
In[99]:= f[x_] := y /. sol[[1]]
In[100]:= f[5]

f[6]
Out[100]= 3 - x
Out[101]= 3 - x
In[102]:= DownValues[f]
Out[102]= {HoldPattern[f[x_]] :&> (y /. sol[[1]])}
(* 注意這裡函數定義的右手邊 x 沒有顯式出現。*)
In[64]:= Attributes[SetDelayed]
Out[64]= {HoldAll, Protected, SequenceHold}
In[104]:= TracePrint[f[5]]
During evaluation of In[104]:=  f[5]
During evaluation of In[104]:=   f
During evaluation of In[104]:=   5
During evaluation of In[104]:=  y/. sol[[1]]
During evaluation of In[104]:=   ReplaceAll
During evaluation of In[104]:=   y
During evaluation of In[104]:=   sol[[1]]
During evaluation of In[104]:=    Part
During evaluation of In[104]:=    sol
During evaluation of In[104]:=    {{y-&>3-x}}
During evaluation of In[104]:=    1
During evaluation of In[104]:=   {{y-&>3-x}}[[1]]
During evaluation of In[104]:=   {y-&>3-x}
During evaluation of In[104]:=  y/. {y-&>3-x}
During evaluation of In[104]:=  3-x

Out[104]= 3 - x

方法二：

...


In[94]:= DownValues[f]
Out[94]= {HoldPattern[f[x_]] :&> (y /. Solve[x + y == 3, y][[1]])}
(* 注意 x 在函數定義中顯示出現了 *)
In[95]:= TracePrint[f[5]]
During evaluation of In[95]:=  f[5]
During evaluation of In[95]:=   f
During evaluation of In[95]:=   5
During evaluation of In[95]:=  y/. Solve[5+y==3,y][[1]]
(* f[5] 求值的第一步即把在函數定義中顯式出現的 x 取值為 x=5；方法一中 x 在定義里沒有顯式出現，所以也沒有被取值為 x=5。 *)
During evaluation of In[95]:=   ReplaceAll
During evaluation of In[95]:=   y
During evaluation of In[95]:=   Solve[5+y==3,y][[1]]
During evaluation of In[95]:=    Part
During evaluation of In[95]:=    Solve[5+y==3,y]
During evaluation of In[95]:=     Solve
During evaluation of In[95]:=     5+y==3
During evaluation of In[95]:=      Equal
During evaluation of In[95]:=      5+y
During evaluation of In[95]:=       Plus
During evaluation of In[95]:=       5
During evaluation of In[95]:=       y
During evaluation of In[95]:=      3
During evaluation of In[95]:=     y
During evaluation of In[95]:=     5+y==3
During evaluation of In[95]:=     5+y==3
During evaluation of In[95]:=     5+y==3
During evaluation of In[95]:=    {{y-&>-2}}
During evaluation of In[95]:=     List
During evaluation of In[95]:=     {y-&>-2}
During evaluation of In[95]:=      List
During evaluation of In[95]:=      y-&>-2
During evaluation of In[95]:=       Rule
During evaluation of In[95]:=       y
During evaluation of In[95]:=       -2
During evaluation of In[95]:=      y-&>-2
During evaluation of In[95]:=       Rule
During evaluation of In[95]:=       y
During evaluation of In[95]:=       -2
During evaluation of In[95]:=     {y-&>-2}
During evaluation of In[95]:=    {{y-&>-2}}
During evaluation of In[95]:=    1
During evaluation of In[95]:=   {{y-&>-2}}[[1]]
During evaluation of In[95]:=   {y-&>-2}
During evaluation of In[95]:=  y/. {y-&>-2}
During evaluation of In[95]:=  -2

Out[95]= -2