Answer :
To determine which statement is correct, we need to check each option based on the properties of the function [tex]\( f(x) \)[/tex] provided in the table. Given [tex]\( f(x) \)[/tex] is defined to be an odd function, we know that [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex]. With this in mind, let's evaluate each option step-by-step.
### Option A: [tex]\( f(0) + f(-0.53) = 0 \)[/tex]
We know from the table that:
[tex]\[ f(0) = 2 \][/tex]
[tex]\[ f(-0.53) = -3 \][/tex]
So, we calculate:
[tex]\[ f(0) + f(-0.53) = 2 + (-3) = -1 \][/tex]
This is not equal to 0, so option A is not true.
### Option B: [tex]\( f(2) = 5 \)[/tex]
Look at the provided table again. We need the value of [tex]\( f(2) \)[/tex]. However, [tex]\( x = 2 \)[/tex] is not provided directly in the table. Therefore, we cannot verify this directly from the provided information. Thus, option B is inconclusive.
### Option C: There is something wrong with the table. Given the table of values, the function can't be odd.
To verify this option, we need to check whether the values in the table comply with the odd function property that [tex]\( f(-x) = -f(x) \)[/tex]:
- For [tex]\( x = -2 \)[/tex],
[tex]\( f(-2) = 5 \)[/tex], therefore [tex]\( f(2) \)[/tex] should be [tex]\( -5 \)[/tex], but since [tex]\( x = 2 \)[/tex] is not provided in the table, we can't verify directly.
- For [tex]\( x = -0.53 \)[/tex],
[tex]\( f(-0.53) = -3 \)[/tex], then [tex]\( f(0.53) \)[/tex] should be [tex]\( 3 \)[/tex], but from the table [tex]\( f(0.53) = 2 \)[/tex]. This value does not match [tex]\( 3 \)[/tex].
This discrepancy indicates inconsistency with the property of odd functions. Therefore, option C is true.
### Option D: [tex]\( f(-1) - f(2) = -10 \)[/tex]
We need the values of [tex]\( f(-1) \)[/tex] and [tex]\( f(2) \)[/tex] from the table:
- [tex]\( f(-1) \)[/tex] is not provided in the table.
- [tex]\( f(2) \)[/tex] is also not directly provided.
Since we don't have these values, we cannot verify this directly. Thus, option D is inconclusive.
### Option E: [tex]\( f(0.35) + f(-0.53) = -1 \)[/tex]
We need the values:
- [tex]\( f(0.35) \)[/tex] is not provided directly in the table.
- However, [tex]\( f(-0.53) = -3 \)[/tex].
Without [tex]\( f(0.35) \)[/tex], we cannot directly verify this option. Thus, option E is inconclusive.
After evaluating the options, the only option we can definitively verify based on the provided information is option C, which states there is something wrong with the table given the function cannot be odd if the provided values conflict with the odd function property. Therefore, the correct answer is:
C. Something is wrong. Given the table of values, the function can't be odd.
### Option A: [tex]\( f(0) + f(-0.53) = 0 \)[/tex]
We know from the table that:
[tex]\[ f(0) = 2 \][/tex]
[tex]\[ f(-0.53) = -3 \][/tex]
So, we calculate:
[tex]\[ f(0) + f(-0.53) = 2 + (-3) = -1 \][/tex]
This is not equal to 0, so option A is not true.
### Option B: [tex]\( f(2) = 5 \)[/tex]
Look at the provided table again. We need the value of [tex]\( f(2) \)[/tex]. However, [tex]\( x = 2 \)[/tex] is not provided directly in the table. Therefore, we cannot verify this directly from the provided information. Thus, option B is inconclusive.
### Option C: There is something wrong with the table. Given the table of values, the function can't be odd.
To verify this option, we need to check whether the values in the table comply with the odd function property that [tex]\( f(-x) = -f(x) \)[/tex]:
- For [tex]\( x = -2 \)[/tex],
[tex]\( f(-2) = 5 \)[/tex], therefore [tex]\( f(2) \)[/tex] should be [tex]\( -5 \)[/tex], but since [tex]\( x = 2 \)[/tex] is not provided in the table, we can't verify directly.
- For [tex]\( x = -0.53 \)[/tex],
[tex]\( f(-0.53) = -3 \)[/tex], then [tex]\( f(0.53) \)[/tex] should be [tex]\( 3 \)[/tex], but from the table [tex]\( f(0.53) = 2 \)[/tex]. This value does not match [tex]\( 3 \)[/tex].
This discrepancy indicates inconsistency with the property of odd functions. Therefore, option C is true.
### Option D: [tex]\( f(-1) - f(2) = -10 \)[/tex]
We need the values of [tex]\( f(-1) \)[/tex] and [tex]\( f(2) \)[/tex] from the table:
- [tex]\( f(-1) \)[/tex] is not provided in the table.
- [tex]\( f(2) \)[/tex] is also not directly provided.
Since we don't have these values, we cannot verify this directly. Thus, option D is inconclusive.
### Option E: [tex]\( f(0.35) + f(-0.53) = -1 \)[/tex]
We need the values:
- [tex]\( f(0.35) \)[/tex] is not provided directly in the table.
- However, [tex]\( f(-0.53) = -3 \)[/tex].
Without [tex]\( f(0.35) \)[/tex], we cannot directly verify this option. Thus, option E is inconclusive.
After evaluating the options, the only option we can definitively verify based on the provided information is option C, which states there is something wrong with the table given the function cannot be odd if the provided values conflict with the odd function property. Therefore, the correct answer is:
C. Something is wrong. Given the table of values, the function can't be odd.
