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已知

f(x), g(x)

都是定义在

\mathbf{R}

上的函数, 对任意

x, y

满足 $f(x-y)=f(x) g(y)-g(x) f(...

Mar 8, 2024

已知

f(x), g(x)

都是定义在

\mathbf{R}

上的函数, 对任意

x, y

满足

f(x-y)=f(x) g(y)-g(x) f(y)

, 且

f(-2)=f(1) \neq 0

, 则下列说法正确的是 ( ) A.

f(0)=1

B. 函数

g(2 x+1)

的图象关于点

(1,0)

对称 C.

g(1)+g(-1)=0

D. 若

f(1)=1

, 则

\sum_{n=1}^{2023} f(n)=1

Solution by Steps

step 2

Substituting

x = y

into the functional equation gives

f(0) = f(x)g(x) - g(x)f(x) = 0

step 3

Since

f(-2) = f(1) \neq 0

f(0)

cannot be 1, thus option A is incorrect..

step 4

To check if

g(2x+1)

is symmetric about the point

(1,0)

, we need to verify if

g(2(1-x)+1) = g(2x+1)

step 5

Since we do not have an explicit form for

g(x)

, we cannot confirm the symmetry without additional information, so option B cannot be confirmed..

step 6

To verify if

g(1) + g(-1) = 0

, we can use the given functional equation by setting

x = 1

and

y = -1

step 7

Substituting

x = 1

and

y = -1

into the functional equation gives

f(1-(-1)) = f(1)g(-1) - g(1)f(-1)

step 8

Since

f(-2) = f(1) \neq 0

, we have

f(2) = f(1)g(-1) - g(1)f(1)

step 9

Simplifying gives

f(2) = f(1)(g(-1) - g(1))

, and since

f(1) \neq 0

, it follows that

g(-1) + g(1) = 0

, so option C is correct..

step 10

To evaluate

\sum_{n=1}^{2023} f(n)

given

f(1) = 1

, we would need more information about the function

f

, so we cannot confirm option D without additional information..

Key Concept

Functional Equation

Explanation

The functional equation

f(x-y) = f(x)g(y) - g(x)f(y)

can be used to find relationships between the values of

f

and

g

at different points.