To solve this problem, let's carefully evaluate each of the given statements based on the provided function values and the property that the function is odd.
A function [tex]\( f(x) \)[/tex] is defined to be odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex]. Let's apply this property and check each statement step-by-step.
1. Given Table of Values:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & -2 & -0.35 & 0 & 0.53 & 1 \\
\hline
$f(x)$ & 5 & -3 & 2 & 2 & -5 \\
\hline
\end{tabular}
\][/tex]
2. Statement A: [tex]\( f(0) + f(-0.53) = 0 \)[/tex]
- From the table: [tex]\( f(0) = 2 \)[/tex]
- Since the function is odd, [tex]\( f(-0.53) = -f(0.53) \)[/tex]
- From the table: [tex]\( f(0.53) = 2 \)[/tex]
- Thus: [tex]\( f(-0.53) = -2 \)[/tex]
Now, let's add these values:
[tex]\[
f(0) + f(-0.53) = 2 + (-2) = 0
\][/tex]
Therefore, statement A is True.
3. Statement B: [tex]\( f(2) = 5 \)[/tex]
- From the table, the value for [tex]\( f(2) \)[/tex] is not directly given.
- Using the odd property: [tex]\( f(-2) = 5 \implies f(2) = -5 \)[/tex] (since [tex]\( f(-x) = -f(x) \)[/tex])
Therefore, statement B is False.
4. Statement C: Something is wrong. Given the table of values, the function can't be odd.
- We need to validate the consistency with the odd function property. We have already checked some values:
- [tex]\( f(-0.53) = -f(0.53) \)[/tex]
- [tex]\( f(2) = -f(-2) \)[/tex]
From the given table, the values are consistent with the odd function property.
Therefore, statement C is False.
5. Statement D: [tex]\( f(-1) - f(2) = -10 \)[/tex]
- From the table: [tex]\( f(1) = -5 \)[/tex]
- Therefore: [tex]\( f(-1) = 5 \)[/tex] (since [tex]\( f(-x) = -f(x) \)[/tex])
- From the odd property and previous result: [tex]\( f(2) = -5 \)[/tex]
- Let's calculate:
[tex]\[
f(-1) - f(2) = 5 - (-5) = 5 + 5 = 10
\][/tex]
Therefore, statement D is False.
6. Statement E: [tex]\( f(0.35) + f(-0.53) = -1 \)[/tex]
- From the table: [tex]\( f(-0.35) = -3 \)[/tex]
- Therefore: [tex]\( f(0.35) = 3 \)[/tex] (since [tex]\( f(-x) = -f(x) \)[/tex])
- We already have: [tex]\( f(-0.53) = -2 \)[/tex]
Let’s add these values:
[tex]\[
f(0.35) + f(-0.53) = 3 + (-2) = 1
\][/tex]
Therefore, statement E is False.
Given our evaluations, only Statement A is true. Thus, the correct answer is:
A: [tex]\( f(0) + f(-0.53) = 0 \)[/tex] is True.
