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Suppose you're given the following table of values for the function [tex]$f(x)$[/tex], and you're told that the function is odd:

\begin{tabular}{|c|c|c|c|c|c|}
\hline [tex]$x$[/tex] & -2 & -0.35 & 0 & 0.53 & 1 \\
\hline [tex]$f(x)$[/tex] & 5 & -3 & 2 & 2 & -5 \\
\hline
\end{tabular}

Then:

A. [tex]$f(0) + f(-0.53) = 0$[/tex]

B. [tex]$f(2) = 5$[/tex]

C. Something is wrong. Given the table of values, the function can't be odd.

D. [tex]$f(-1) - f(2) = -10$[/tex]

E. [tex]$f(0.35) + f(-0.53) = -1$[/tex]

Sagot :

GinnyAnswer
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.
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