Let's analyze and verify each statement given the function [tex]\(f(x) = 3(2.5)^x\)[/tex].
1. The function is exponential.
Yes, the function [tex]\(f(x) = 3(2.5)^x\)[/tex] is an exponential function because it is of the form [tex]\(a \cdot b^x\)[/tex] where [tex]\(a\)[/tex] is a constant (in this case, [tex]\(3\)[/tex]) and [tex]\(b\)[/tex] is a positive constant base (in this case, [tex]\(2.5\)[/tex]). Exponential functions have the characteristic form where the variable [tex]\(x\)[/tex] is in the exponent.
True.
2. The initial value of the function is 2.5.
The initial value of the function is determined by evaluating [tex]\(f(x)\)[/tex] at [tex]\(x = 0\)[/tex].
[tex]\[
f(0) = 3(2.5)^0 = 3 \cdot 1 = 3
\][/tex]
So, the initial value of the function is 3, not 2.5.
False.
3. The function increases by a factor of 2.5 for each unit increase in [tex]\(x\)[/tex].
For exponential functions of the form [tex]\(a \cdot b^x\)[/tex], as [tex]\(x\)[/tex] increases by 1, the value of the function is multiplied by the base [tex]\(b\)[/tex]. Here, the base [tex]\(b\)[/tex] is 2.5. Thus, for each unit increase in [tex]\(x\)[/tex], [tex]\(f(x)\)[/tex] increases by a factor of 2.5.
True.
4. The domain of the function is all real numbers.
For an exponential function [tex]\(a \cdot b^x\)[/tex], the domain is all real numbers because you can input any real number for [tex]\(x\)[/tex] and get a valid output.
True.
5. The range of the function is all real numbers greater than 3.
The function [tex]\(f(x) = 3(2.5)^x\)[/tex] has a range that includes all positive real numbers. It approaches 0 as [tex]\(x\)[/tex] approaches negative infinity but never actually reaches 0. Since the coefficient 3 does not shift the range upwards, the range is all positive real numbers (greater than 0), not necessarily greater than 3.
False.
Based on the evaluation, the true statements are:
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.
So the correct answers are:
[tex]\((\text{True}, \text{False}, \text{True}, \text{True}, \text{False})\)[/tex].
