To solve the problem, let's carefully analyze the given task and provide a detailed explanation step by step.
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Problem Statement
The task involves solving a mathematical problem related to functions and their properties. The specific question is:
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Question 1: Let \( f(x) \) be a function defined on the interval \([-2, 2]\). Suppose that \( f(x) \) satisfies the following conditions:
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> 1. \( f(x) \) is an even function.
> 2. \( f(x) \) is increasing on the interval \([0, 2]\).
> 3. \( f(0) = 0 \).
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> Determine whether the following statements are true or false:
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> (a) \( f(-1) < f(1) \)
> (b) \( f(-2) > f(1) \)
> (c) \( f(-1) = f(1) \)
> (d) \( f(2) > f(0) \)
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Step-by-Step Solution
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Step 1: Understand the properties of the function \( f(x) \)
1.
Even Function: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in its domain. This means the graph of \( f(x) \) is symmetric about the \( y \)-axis.
2.
Increasing on \([0, 2]\): The function \( f(x) \) is increasing on the interval \([0, 2]\), which means for any \( x_1, x_2 \in [0, 2] \) with \( x_1 < x_2 \), we have \( f(x_1) < f(x_2) \).
3.
\( f(0) = 0 \): The function value at \( x = 0 \) is given as \( f(0) = 0 \).
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Step 2: Analyze each statement
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(a) \( f(-1) < f(1) \)
- Since \( f(x) \) is an even function, we know \( f(-1) = f(1) \).
- Therefore, the statement \( f(-1) < f(1) \) is
false because \( f(-1) \) is equal to \( f(1) \), not less than it.
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(b) \( f(-2) > f(1) \)
- Using the property of even functions, \( f(-2) = f(2) \).
- Since \( f(x) \) is increasing on \([0, 2]\), we know \( f(1) < f(2) \).
- Therefore, \( f(-2) = f(2) > f(1) \).
- The statement \( f(-2) > f(1) \) is
true.
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(c) \( f(-1) = f(1) \)
- By the definition of an even function, \( f(-1) = f(1) \).
- This statement is
true.
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(d) \( f(2) > f(0) \)
- Since \( f(x) \) is increasing on \([0, 2]\), for any \( x_1, x_2 \in [0, 2] \) with \( x_1 < x_2 \), we have \( f(x_1) < f(x_2) \).
- Here, \( 0 < 2 \), so \( f(0) < f(2) \).
- Given \( f(0) = 0 \), it follows that \( f(2) > 0 \).
- Therefore, the statement \( f(2) > f(0) \) is
true.
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Final Answer
Summarizing the results for each statement:
- (a) False
- (b) True
- (c) True
- (d) True
Thus, the final answer is:
\[
\boxed{\text{(a) False, (b) True, (c) True, (d) True}}
\]
Parent Tip: Review the logic above to help your child master the concept of mitosis versus meiosis worksheet answer key.
