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Sagot :
Given the information about the function [tex]\( f \)[/tex], let's analyze each statement one by one.
1. The domain of [tex]\( f(x) \)[/tex] is \{all real numbers\}:
- Without specific information about the function [tex]\( f \)[/tex], it's impossible to conclusively determine its domain. Therefore, we cannot assert that the domain of [tex]\( f(x) \)[/tex] is all real numbers.
2. The range of [tex]\( f(x) \)[/tex] is \{all real numbers\}:
- Similarly, without knowing more about [tex]\( f \)[/tex], we cannot determine if it can take every real number as an output. Therefore, it's not possible to confirm that the range of [tex]\( f(x) \)[/tex] is all real numbers.
3. The domain of [tex]\( f(x) \)[/tex] is [tex]\(\{x \mid x>0\}\)[/tex]:
- This statement implies that [tex]\( f(x) \)[/tex] is defined only for positive values of [tex]\( x \)[/tex]. Again, without knowing about the function's definition, it is impossible to affirm this statement.
4. The range of [tex]\( f(x) \)[/tex] is [tex]\(\{y \mid y>0\}\)[/tex]:
- This statement suggests that [tex]\( f(x) \)[/tex] only produces positive output values [tex]\( y \)[/tex]. With insufficient details, we cannot validate or refute this claim either.
5. [tex]\( f(2)=4 \)[/tex]:
- This asserts a specific output value for [tex]\( f \)[/tex] when the input is 2. Without knowing the form of [tex]\( f \)[/tex], we cannot confirm this statement.
6. [tex]\( f(2)=2\left(\frac{1}{2} x + 3\right) \)[/tex]:
- This suggests an equation involving [tex]\( f(2) \)[/tex]. Simplifying the expression inside,
[tex]\[ f(2) = 2\left(\frac{1}{2}(2) + 3\right) = 2(1 + 3) = 2 \cdot 4 = 8 \][/tex]
- Therefore, this equation states [tex]\( f(2) = 8 \)[/tex]. Without the definition of [tex]\( f \)[/tex], it is unclear if this assertion holds true, or if the function's value would indeed be 8 for [tex]\( x = 2 \)[/tex].
Given these analyses, none of the statements about [tex]\( f \)[/tex] can be verified as true without additional information on the function [tex]\( f \)[/tex]. Therefore, the answer to which statements about the function [tex]\( f \)[/tex] are true is:
```
None
```
1. The domain of [tex]\( f(x) \)[/tex] is \{all real numbers\}:
- Without specific information about the function [tex]\( f \)[/tex], it's impossible to conclusively determine its domain. Therefore, we cannot assert that the domain of [tex]\( f(x) \)[/tex] is all real numbers.
2. The range of [tex]\( f(x) \)[/tex] is \{all real numbers\}:
- Similarly, without knowing more about [tex]\( f \)[/tex], we cannot determine if it can take every real number as an output. Therefore, it's not possible to confirm that the range of [tex]\( f(x) \)[/tex] is all real numbers.
3. The domain of [tex]\( f(x) \)[/tex] is [tex]\(\{x \mid x>0\}\)[/tex]:
- This statement implies that [tex]\( f(x) \)[/tex] is defined only for positive values of [tex]\( x \)[/tex]. Again, without knowing about the function's definition, it is impossible to affirm this statement.
4. The range of [tex]\( f(x) \)[/tex] is [tex]\(\{y \mid y>0\}\)[/tex]:
- This statement suggests that [tex]\( f(x) \)[/tex] only produces positive output values [tex]\( y \)[/tex]. With insufficient details, we cannot validate or refute this claim either.
5. [tex]\( f(2)=4 \)[/tex]:
- This asserts a specific output value for [tex]\( f \)[/tex] when the input is 2. Without knowing the form of [tex]\( f \)[/tex], we cannot confirm this statement.
6. [tex]\( f(2)=2\left(\frac{1}{2} x + 3\right) \)[/tex]:
- This suggests an equation involving [tex]\( f(2) \)[/tex]. Simplifying the expression inside,
[tex]\[ f(2) = 2\left(\frac{1}{2}(2) + 3\right) = 2(1 + 3) = 2 \cdot 4 = 8 \][/tex]
- Therefore, this equation states [tex]\( f(2) = 8 \)[/tex]. Without the definition of [tex]\( f \)[/tex], it is unclear if this assertion holds true, or if the function's value would indeed be 8 for [tex]\( x = 2 \)[/tex].
Given these analyses, none of the statements about [tex]\( f \)[/tex] can be verified as true without additional information on the function [tex]\( f \)[/tex]. Therefore, the answer to which statements about the function [tex]\( f \)[/tex] are true is:
```
None
```
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