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已知连续函数 f(x)f(x) 满足: (1) ∀x,y∈R\forall x, y \in \mathrm{R}, 则有 f(x+y)=f(x)+f(y)−1f(x+y)=f(x)+f(y)-1, (2...
Mar 8, 2024
已知连续函数 f(x)f(x) 满足: (1) ∀x,y∈R\forall x, y \in \mathrm{R}, 则有 f(x+y)=f(x)+f(y)−1f(x+y)=f(x)+f(y)-1, (2)当 x>0 时， f(x)<1, (3) f(1)=−2f(1)=-2, 则不等式 f\left(3 x^{2}\right)-2 f(x)>f(3 x)+4 的解集为
Solution by Steps
step 1
Given the functional equation f(x+y)=f(x)+f(y)−1 f(x+y) = f(x) + f(y) - 1 , we can find f(0) f(0) by setting x=y=0 x = y = 0
step 2
Substituting into the equation, we get f(0+0)=f(0)+f(0)−1 f(0+0) = f(0) + f(0) - 1 , which simplifies to f(0)=12 f(0) = \frac{1}{2}
step 3
Using f(1)=−2 f(1) = -2 and the functional equation, we can find f(2) f(2) by setting x=y=1 x = y = 1
step 4
We have f(2)=f(1+1)=f(1)+f(1)−1=−2−2−1=−5 f(2) = f(1+1) = f(1) + f(1) - 1 = -2 - 2 - 1 = -5
step 5
To solve the inequality f(3x^2) - 2f(x) > f(3x) + 4 , we substitute f(0)=12 f(0) = \frac{1}{2} and f(2)=−5 f(2) = -5 into the inequality
step 6
We need to consider the cases when x=0 x = 0 and x≠0 x \neq 0 separately
step 7
For x=0 x = 0 , the inequality simplifies to f(0) - 2f(0) > f(0) + 4 , which is not true since 12−2⋅12 \frac{1}{2} - 2 \cdot \frac{1}{2} is not greater than 12+4 \frac{1}{2} + 4
step 8
For x≠0 x \neq 0 , we need to find the values of x x for which the inequality holds true
step 9
Since f(x) < 1 for x > 0 , we can deduce that f(3x^2) < 1 and f(3x) < 1 for x > 0
step 10
The inequality f(3x^2) - 2f(x) > f(3x) + 4 can be rewritten as f(3x^2) - f(3x) > 2f(x) + 4
step 11
Since f(3x^2) < 1 and f(3x) < 1 , the left side of the inequality is less than 1−1=0 1 - 1 = 0
step 12
Therefore, the inequality cannot hold for any x > 0 because the right side 2f(x)+4 2f(x) + 4 is always greater than 4
step 13
The solution set for the inequality is the empty set, ∅ \emptyset , since no values of x x satisfy the inequality
Answer
The solution set for the inequality f(3x^2) - 2f(x) > f(3x) + 4 is ∅ \emptyset .
Key Concept
Functional Equations and Inequalities
Explanation
The solution involves using the given functional equation to find specific function values and then applying these values to analyze the inequality. The inequality cannot be satisfied for any x > 0 due to the constraints on f(x) f(x) .
Please note that the steps provided are based on the constraints given in the problem and the properties of the function f(x) f(x) . The actual steps may vary depending on the specific properties of the function, which are not fully detailed in the question.
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