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現在有函數f(x)=-x+4

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且其曲線已經在坐標平面上畫出來了

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我們試著求一下其反函數

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要求反函數

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我常做的是設變量y

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y=f(x) 或者寫成

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y=-x+4

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現在 我們用x表示了y

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爲了求反函數 要反過來

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用y表示x

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兩邊同時減4

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得到y-4=-x

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要求出x

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可以對方程兩邊

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同時乘以-1

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得到-y+4=x

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因爲我們習慣於

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把自變量寫在左邊

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因此可以改寫成x=-y+4

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還有另一種寫法

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就是f^(-1) (y)=-y+4

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這個就是反函數

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我們把它寫成了y的函數

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爲了得到x的函數 我們可以把y命名爲x

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我們來做一下

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把y重命名爲x

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得到f^(-1) (x)=-x+4

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這兩個函數是等價的

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在這裡我們用y表示自變量

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或者說是輸入變量

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在這兒則是用x 不過這兩者是完全一樣的

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現在 爲了有趣 我們畫出反函數的曲線

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看看它和這條曲線之間的聯係

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如果看這個函數 它和原函數看起來完全一樣

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都是-x+4

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是同一個函數

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我們看一下 如果我們-- y的截距是4

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這兩條曲線應該是一樣的

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這函數與自己互成反函數

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如果要畫出來

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應該把它畫到這條線上

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有幾種方法思考這一情況

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在第一個反函數的影片裏

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我講過原函數和反函數是--

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它們是關於y=x對稱的

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那麽曲線y=x在哪呢？

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y=x是這樣子的

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而y=-x+4實際上是垂直於

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y=x的 所以如果取對稱

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實際上就是把它翻過來

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是同一條曲線

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自己是自己的映射

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現在我們來確保這是正確的

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當我們討論這個函數時

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如果代入2 會由函數映射成2

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代入4 得到0

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如果反過來會怎樣？

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輸入是2

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兩種方向輸出都是2 這樣可以講得通

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對於原函數 4被映射成0

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對於反函數 0被映射成4

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所以這是完全正確的

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換種方式思考

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對於原函數-- 我把它寫下來

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你們可能對於這很熟悉了 不過僅僅是以防萬一

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寫出來可能會有幫助的

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我們選f(5)

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f(5)=-1

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或者說原函數把5映射成-1

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那麽反函數呢？

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f^(-1) (-1)是多少呢？

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f^(-1) (-1)=5

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或者可以說它把-1映射到5

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如果你們想到了集合的概念

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也就是定義域和值域

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假設這是f的定義域

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這是f的值域

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f會從5得到-1

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這就是f的作用

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同時我們知道f^(-1)從-1回到5

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f^(-1)把-1變回5

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這也是我們所期望的

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我再做一道

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已知g(x)=-2x-1

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就像上個問題 設y等於它

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y=g(x)

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也就等於-2x-1

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現在要求x

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y+1=-2x

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這一步是兩邊同時加1

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現在方程兩邊同時除以-2

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得到(-y)/2-1/2=x

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或者寫成x=(-y)/2-1/2

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或者寫成

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f^(-1) (y)=(-y)/2-1/2

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我們直接把y命名爲x

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也就有--

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我要仔細點了 這不是f

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原函數是g 我得說清楚這點

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應該是g^(-1) (y)=(-y)/2-1/2

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因爲是以g(x)作爲開始的

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不是f(x)

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要確保用對符號

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我們可以重命名y並得到

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g^(-1) (x)=(-x)/2-1/2

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現在來畫一下圖

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y截距是-1/2

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這個點在那

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斜率是-1/2

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如果從-1/2開始

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沿正方向移1

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會下降1/2

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如果再移動1個單位 縱坐標又會下降1/2

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如果沿反方向移動-- 會變成這樣

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我盡最大努力來畫

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曲線應該是這樣子的

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它會一直延伸 所以應該是這樣子

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它會沿兩個方向一直延續

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現在我們來看一下它們是否

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關於y=x對稱 y=x是這條曲線

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你們可以看出來 它們確實是對稱的

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如果把這條藍色的曲線沿y=x翻轉

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會得到這條橙色的曲線

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按照字面來理解 反函數的中心思想是

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函數最初被表示爲--

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最初是用x表示y的

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你們要通過做一些變換

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把x用y來表示

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得到的就是以y爲自變量的

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反函數