To analyze the function [tex]\( f(x) = 3(2.5)^x \)[/tex], let's go through each statement step by step:
1. The function is exponential.
- An exponential function is of the form [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] is a constant and [tex]\( b \)[/tex] is the base of the exponent that is positive and not equal to 1.
- Here, [tex]\( f(x) = 3(2.5)^x \)[/tex] fits this form with [tex]\( a = 3 \)[/tex] and [tex]\( b = 2.5 \)[/tex]. Therefore, this is indeed an exponential function.
- True
2. The initial value of the function is 2.5.
- The initial value of a function is the value when [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into the function, we get [tex]\( f(0) = 3(2.5)^0 = 3 \cdot 1 = 3 \)[/tex].
- Therefore, the initial value is 3, not 2.5.
- False
3. The function increases by a factor of 2.5 for each unit increase in [tex]\( x \)[/tex].
- This means that for every 1-unit increase in [tex]\( x \)[/tex], the value of the function is multiplied by 2.5.
- This is a characteristic of the exponential function [tex]\( f(x) = 3(2.5)^x \)[/tex]. Therefore, it is true.
- True
4. The domain of the function is all real numbers.
- The domain of an exponential function is all real numbers because you can substitute any real number into the exponent and still have a valid output value.
- Therefore, the domain of [tex]\( f(x) = 3(2.5)^x \)[/tex] is all real numbers.
- True
5. The range of the function is all real numbers greater than 3.
- The range of an exponential function [tex]\( a \cdot b^x \)[/tex] where [tex]\( a > 0 \)[/tex] and [tex]\( b > 1 \)[/tex] is all positive real numbers greater than 0.
- For [tex]\( f(x) = 3(2.5)^x \)[/tex], as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches 0 but never reaches 0. It is always positive.
- The range is all real numbers greater than 0, not greater than 3.
- False
Putting it all together, here are the true statements about the function [tex]\( f(x) = 3(2.5)^x \)[/tex]:
1. The function is exponential.
2. The function increases by a factor of 2.5 for each unit increase in [tex]\( x \)[/tex].
3. The domain of the function is all real numbers.
And here are the false statements:
1. The initial value of the function is 2.5.
2. The range of the function is all real numbers greater than 3.
Hence, the result is: `(True, False, True, True, False)`.
