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通过复合函数验证反函数

了解如何通过函数复合来验证两个函数是否互逆。例如, f (x) = 5 x-7 和 g (x) = x/5 + 7 互为反函数吗?
本文涵盖了许多函数组合。如果你需要复习这一课题,我们建议你在阅读本文之前,点击这里
反函数,最概括来讲,就是“相反”的两个函数。比如,如果一个函数里,a 可以得出 b,那么它的反函数就是通过 b 得出 a
我们以函数 f 和函数 g 为例:f f, left parenthesis, x, right parenthesis, equals, start fraction, x, plus, 1, divided by, 3, end fraction 以及g, left parenthesis, x, right parenthesis, equals, 3, x, minus, 1
注意: f, left parenthesis, 5, right parenthesis, equals, 2g, left parenthesis, 2, right parenthesis, equals, 5
我们看到,当我们求f 之后再求 g,就得出了原来的自变量。表示成组合函数,就是 g, left parenthesis, f, left parenthesis, 5, right parenthesis, right parenthesis, equals, 5
但是两个函数要互为反函数,我们必须证明这适用于任何可能的自变量 ,无论代入 fg 的顺序如何。这就是反函数组合原则。

反函数组合原则

这是两个函数 fg 互为反函数的条件:
  • f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, equals, x ,在定义域g区间内任何x 都成立
  • g, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis, equals, x ,在定义域f区间内任何 x 都成立
这是因为如果 fg 互为反函数,将 fg 组合起来(两种方式均可)得出的函数,就会使每个自变量都得出该自变量本身。我们称此函数为”恒等函数“。

例1:函数 fg 互为反函数

我们使用发函数组合原则来验证上述函数 fg 确实互为反函数。
要记得, f, left parenthesis, x, right parenthesis, equals, start fraction, x, plus, 1, divided by, 3, end fractiong, left parenthesis, x, right parenthesis, equals, 3, x, minus, 1
我们求 f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesisg, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis
f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesisg, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis
f(g(x))=g(x)+13=3x1+13=3x3=x\begin{aligned} f(\greenD{g(x)})&=\dfrac{\greenD{g(x)}+1}{3}\\\\&=\dfrac{\greenD{3x-1}+1}{3}\\\\&=\dfrac{3x}{3}\\\\&=x\\\end{aligned}g(f(x))=3(f(x))1=3(x+13)1=x+11=x\qquad\qquad \begin{aligned}g(\purpleC{f(x)})&=3\left(\purpleC{f(x)}\right)-1\\\\&=3\left(\purpleC{\dfrac{x+1}{3}}\right)-1\\\\&=x+1-1\\\\&=x\\\end{aligned}
我们看到函数 fg 互为相反,因为f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, equals, xg, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis, equals, x

例2:函数 fg 不是互为反函数

如果f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesisg, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis不等于 x,那么 fg 不会是互为反函数。
我们试试 f, left parenthesis, x, right parenthesis, equals, 5, x, minus, 7g, left parenthesis, x, right parenthesis, equals, start fraction, x, divided by, 5, end fraction, plus, 7
f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesisg, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis
f(g(x))=5(g(x))7=5(x5+7)7=x+357=x+28\begin{aligned} f(\greenD{g(x)})&=5(\greenD{g(x)})-7\\\\&=5\left(\greenD{\dfrac{x}{5}+7}\right)-7\\\\&=x+35-7\\\\&=x+28\end{aligned}\qquad g(f(x))=f(x)5+7=5x75+7=x75+7=x+285\qquad\begin{aligned} g(\purpleC{f(x)})&=\dfrac{\purpleC{f(x)}}{5}+7\\\\&=\dfrac{\purpleC{5x-7}}{5}+7\\\\&=x-\dfrac75+7\\\\&=x+\dfrac{28}{5}\\\end{aligned}
所以函数 fg 不是互为反函数,因为 f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, does not equal, xg, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis, does not equal, x
(这里要注意,看到 f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, equals, x, plus, 28,我们可以得出结论,就是 fg 不是互为反函数。)

检查你对内容的理解

总的来讲,如果 fg 互为反函数,我们可以把它们组合起来。如果结果是 x,这两个函数就是互为反函数。否则则不是。

1) f, left parenthesis, x, right parenthesis, equals, 2, x, plus, 7h, left parenthesis, x, right parenthesis, equals, start fraction, x, minus, 7, divided by, 2, end fraction

f, left parenthesis, h, left parenthesis, x, right parenthesis, right parenthesish, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis 表达成变量为 x的函数形式。
f, left parenthesis, h, left parenthesis, x, right parenthesis, right parenthesis, equals
h, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis, equals
函数 fh 是否互为反函数?
选出正确答案:
选出正确答案:

2) f, left parenthesis, x, right parenthesis, equals, 4, x, plus, 10g, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, 4, end fraction, x, minus, 10

用简单的方式表达 f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesisg, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis ,以x为变量。
f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, equals
g, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis, equals
函数 fg 是否互为反函数?
选出正确答案:
选出正确答案:

3) f, left parenthesis, x, right parenthesis, equals, start fraction, 2, divided by, 3, end fraction, x, minus, 8h, left parenthesis, x, right parenthesis, equals, start fraction, 3, divided by, 2, end fraction, left parenthesis, x, plus, 8, right parenthesis

f, left parenthesis, h, left parenthesis, x, right parenthesis, right parenthesish, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis 表达成变量为 x的函数形式。
f, left parenthesis, h, left parenthesis, x, right parenthesis, right parenthesis, equals
h, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis, equals
函数 fh 是否互为反函数?
选出正确答案:
选出正确答案: