AMC12 2004 B

AMC12 2004 B · Q13

AMC12 2004 B · Q13. It mainly tests Linear equations, Functions basics.

If $f(x) = ax+b$ and $f^{-1}(x) = bx+a$ with $a$ and $b$ real, what is the value of $a+b$?

若$f(x) = ax+b$且$f^{-1}(x) = bx+a$，其中$a$和$b$为实数，求$a+b$的值。

(A) -2 -2

(B) -1 -1

(D) 1 1

(E) 2 2

Answer

Correct choice: (A)

正确答案：（A）

Solution

Since $f(f^{-1}(x))=x$, it follows that $a(bx+a)+b=x$, which implies $abx + a^2 +b = x$. This equation holds for all values of $x$ only if $ab=1$ and $a^2+b=0$. Then $b = -a^2$. Substituting into the equation $ab = 1$, we get $-a^3 = 1$. Then $a = -1$, so $b = -1$, and\[f(x)=-x-1.\]Likewise\[f^{-1}(x)=-x-1.\]These are inverses to one another since\[f(f^{-1}(x))=-(-x-1)-1=x+1-1=x.\]\[f^{-1}(f(x))=-(-x-1)-1=x+1-1=x.\]Therefore $a+b=\boxed{\mathrm{(A)}\ -2}$. note: you could just find that $a=-1$ and $b=-1$ and add, getting $-2$.

由于$f(f^{-1}(x))=x$，可得$a(bx+a)+b=x$，即$abx + a^2 +b = x$。该等式对所有$x$成立当且仅当$ab=1$且$a^2+b=0$。于是$b = -a^2$。代入$ab = 1$得$-a^3 = 1$，因此$a = -1$，从而$b = -1$，并且 \[f(x)=-x-1.\]同样 \[f^{-1}(x)=-x-1.\]它们互为反函数，因为 \[f(f^{-1}(x))=-(-x-1)-1=x+1-1=x.\] \[f^{-1}(f(x))=-(-x-1)-1=x+1-1=x.\] 因此$a+b=\boxed{\mathrm{(A)}\ -2}$。注：也可以直接求出$a=-1$且$b=-1$，相加得到$-2$。

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