Answer :
To analyze the given table and compare the characteristics of the functions, let's break down each statement:
1. [tex]$f(x)$[/tex] has an all negative domain.
- Explanation: The domain here refers to the set of all input values [tex]\( x \)[/tex]. From the table, the domain values are [tex]\(-2, -1, 1, 2\)[/tex]. There are positive numbers in this domain ([tex]\(1\)[/tex] and [tex]\(2\)[/tex]), hence [tex]\(f(x)\)[/tex] does not have an all negative domain.
- Conclusion: This statement is false.
2. [tex]$g(x)$[/tex] has the greatest maximum value.
- Explanation: To determine the maximum value of each function:
- Maximum of [tex]\(f(x)\)[/tex] = 6
- Maximum of [tex]\(g(x)\)[/tex] = 8
- Maximum of [tex]\(h(x)\)[/tex], considering the given values, is [tex]\(-1.5\)[/tex]
- [tex]\(g(x)\)[/tex] indeed has the highest maximum value among the functions.
- Conclusion: This statement is true.
3. All three functions share the same range.
- Explanation: The ranges of the functions, from the table, are:
- Range of [tex]\(f(x)\)[/tex]: \{4, 4.5, 5.5, 6\}
- Range of [tex]\(g(x)\)[/tex]: \{6, 6.5, 7.5, 8\}
- Range of [tex]\(h(x)\)[/tex]: \{-3, -2.5, -1.5\}
- The ranges are different for each function.
- Conclusion: This statement is false.
4. [tex]$h(x)$[/tex] has a range of all negative numbers.
- Explanation: Using the given values:
- The range of [tex]\(h(x)\)[/tex] is \{-3, -2.5, -1.5\}, which are all negative numbers.
- Conclusion: This statement is true.
5. All three functions share the same domain.
- Explanation: The domain for all three functions is indeed the set of [tex]\( x \)[/tex] values presented in the table, which are [tex]\(-2, -1, 1, 2\)[/tex].
- Conclusion: This statement is true.
From these observations, the two correct statements that can be used to compare the characteristics of the functions are:
4. [tex]$h(x)$[/tex] has a range of all negative numbers.
5. All three functions share the same domain.
Thus, the selected options are:
- [tex]$h(x)$[/tex] has a range of all negative numbers.
- All three functions share the same domain.
1. [tex]$f(x)$[/tex] has an all negative domain.
- Explanation: The domain here refers to the set of all input values [tex]\( x \)[/tex]. From the table, the domain values are [tex]\(-2, -1, 1, 2\)[/tex]. There are positive numbers in this domain ([tex]\(1\)[/tex] and [tex]\(2\)[/tex]), hence [tex]\(f(x)\)[/tex] does not have an all negative domain.
- Conclusion: This statement is false.
2. [tex]$g(x)$[/tex] has the greatest maximum value.
- Explanation: To determine the maximum value of each function:
- Maximum of [tex]\(f(x)\)[/tex] = 6
- Maximum of [tex]\(g(x)\)[/tex] = 8
- Maximum of [tex]\(h(x)\)[/tex], considering the given values, is [tex]\(-1.5\)[/tex]
- [tex]\(g(x)\)[/tex] indeed has the highest maximum value among the functions.
- Conclusion: This statement is true.
3. All three functions share the same range.
- Explanation: The ranges of the functions, from the table, are:
- Range of [tex]\(f(x)\)[/tex]: \{4, 4.5, 5.5, 6\}
- Range of [tex]\(g(x)\)[/tex]: \{6, 6.5, 7.5, 8\}
- Range of [tex]\(h(x)\)[/tex]: \{-3, -2.5, -1.5\}
- The ranges are different for each function.
- Conclusion: This statement is false.
4. [tex]$h(x)$[/tex] has a range of all negative numbers.
- Explanation: Using the given values:
- The range of [tex]\(h(x)\)[/tex] is \{-3, -2.5, -1.5\}, which are all negative numbers.
- Conclusion: This statement is true.
5. All three functions share the same domain.
- Explanation: The domain for all three functions is indeed the set of [tex]\( x \)[/tex] values presented in the table, which are [tex]\(-2, -1, 1, 2\)[/tex].
- Conclusion: This statement is true.
From these observations, the two correct statements that can be used to compare the characteristics of the functions are:
4. [tex]$h(x)$[/tex] has a range of all negative numbers.
5. All three functions share the same domain.
Thus, the selected options are:
- [tex]$h(x)$[/tex] has a range of all negative numbers.
- All three functions share the same domain.
