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Sagot :
Let's analyze the given function [tex]\( f(x) = 3(2.5)^x \)[/tex] and determine which of the given statements are true.
1. The function is exponential.
- A function of the form [tex]\( f(x) = a(b)^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants and [tex]\( b > 0 \)[/tex], is an exponential function.
- Here, [tex]\( f(x) = 3(2.5)^x \)[/tex] fits this form with [tex]\( a = 3 \)[/tex] and [tex]\( b = 2.5 \)[/tex], so the statement is true.
2. The initial value of the function is 2.5.
- The initial value of the function is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
- Evaluate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 3(2.5)^0 = 3 \cdot 1 = 3 \][/tex]
- Therefore, the initial value is 3, not 2.5. This statement is false.
3. The function increases by a factor of 2.5 for each unit increase in [tex]\( x \)[/tex].
- For an exponential function [tex]\( f(x) = a(b)^x \)[/tex], the function increases by a factor of [tex]\( b \)[/tex] for each unit increase in [tex]\( x \)[/tex].
- Here, [tex]\( b = 2.5 \)[/tex], so the function increases by a factor of 2.5 for each unit increase in [tex]\( x \)[/tex]. This statement is true.
4. The domain of the function is all real numbers.
- The domain of an exponential function [tex]\( f(x) = a(b)^x \)[/tex] is all real numbers, since you can plug any real number into [tex]\( x \)[/tex] and the expression is defined.
- Therefore, the domain of [tex]\( f(x) = 3(2.5)^x \)[/tex] is all real numbers. This statement is true.
5. The range of the function is all real numbers greater than 3.
- The range of an exponential function [tex]\( f(x) = a(b)^x \)[/tex] where [tex]\( a > 0 \)[/tex] and [tex]\( b > 1 \)[/tex] is all positive real numbers greater than zero.
- In this case, [tex]\( f(x) = 3(2.5)^x \)[/tex], which starts from 3 when [tex]\( x = 0 \)[/tex] and increases without bound as [tex]\( x \)[/tex] increases.
- The range is not restricted to being greater than 3, it's greater than 0. This statement is false.
Thus, the true statements are:
- The function is exponential (True).
- The function increases by a factor of 2.5 for each unit increase in [tex]\( x \)[/tex] (True).
- The domain of the function is all real numbers (True).
The false statements are:
- The initial value of the function is 2.5 (False).
- The range of the function is all real numbers greater than 3 (False).
Hence, the result is:
[tex]\[ (\text{True}, \text{False}, \text{True}, \text{True}, \text{False}) \][/tex]
1. The function is exponential.
- A function of the form [tex]\( f(x) = a(b)^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants and [tex]\( b > 0 \)[/tex], is an exponential function.
- Here, [tex]\( f(x) = 3(2.5)^x \)[/tex] fits this form with [tex]\( a = 3 \)[/tex] and [tex]\( b = 2.5 \)[/tex], so the statement is true.
2. The initial value of the function is 2.5.
- The initial value of the function is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
- Evaluate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 3(2.5)^0 = 3 \cdot 1 = 3 \][/tex]
- Therefore, the initial value is 3, not 2.5. This statement is false.
3. The function increases by a factor of 2.5 for each unit increase in [tex]\( x \)[/tex].
- For an exponential function [tex]\( f(x) = a(b)^x \)[/tex], the function increases by a factor of [tex]\( b \)[/tex] for each unit increase in [tex]\( x \)[/tex].
- Here, [tex]\( b = 2.5 \)[/tex], so the function increases by a factor of 2.5 for each unit increase in [tex]\( x \)[/tex]. This statement is true.
4. The domain of the function is all real numbers.
- The domain of an exponential function [tex]\( f(x) = a(b)^x \)[/tex] is all real numbers, since you can plug any real number into [tex]\( x \)[/tex] and the expression is defined.
- Therefore, the domain of [tex]\( f(x) = 3(2.5)^x \)[/tex] is all real numbers. This statement is true.
5. The range of the function is all real numbers greater than 3.
- The range of an exponential function [tex]\( f(x) = a(b)^x \)[/tex] where [tex]\( a > 0 \)[/tex] and [tex]\( b > 1 \)[/tex] is all positive real numbers greater than zero.
- In this case, [tex]\( f(x) = 3(2.5)^x \)[/tex], which starts from 3 when [tex]\( x = 0 \)[/tex] and increases without bound as [tex]\( x \)[/tex] increases.
- The range is not restricted to being greater than 3, it's greater than 0. This statement is false.
Thus, the true statements are:
- The function is exponential (True).
- The function increases by a factor of 2.5 for each unit increase in [tex]\( x \)[/tex] (True).
- The domain of the function is all real numbers (True).
The false statements are:
- The initial value of the function is 2.5 (False).
- The range of the function is all real numbers greater than 3 (False).
Hence, the result is:
[tex]\[ (\text{True}, \text{False}, \text{True}, \text{True}, \text{False}) \][/tex]
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