At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
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]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
