1952 AMC 12 第 11 题

Question

若

y=f(x)=\dfrac{x+2}{x-1}

，则下列说法不正确的是：

If

y=f(x)=\dfrac{x+2}{x-1},

then it is incorrect to say:

x=\dfrac{y+2}{y-1}

f(0)=-2

f(1)=0

f(-2)=0

f(y)=x

代入求值前，先检查有理函数的定义域

Check the domain of the rational function before evaluating it

分母在某个给出的输入值处为零

The denominator vanishes at one of the listed inputs

该函数在

x=1

处无定义，因为此时分母为零，所以

f(1)=0

是错误的。其他说法均成立：

f(0)=-2

、

f(-2)=0

；解方程

y=\frac{x+2}{x-1}

得

x=\frac{y+2}{y-1}

，同一公式也说明

f

是自身的反函数。

因此，正确答案是 C。

The function is undefined at

x=1

because its denominator is zero there, so

f(1)=0

is false. The other statements hold:

f(0)=-2,

f(-2)=0,

solving

y=\frac{x+2}{x-1}

gives

x=\frac{y+2}{y-1},

and this same formula shows that

f

is its own inverse.

Thus, the correct answer is C.

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