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Which statements are true of the function [tex]f(x)=3(2.5)^x[/tex]? Check all that apply.

A. The function is exponential.
B. The initial value of the function is 2.5.
C. The function increases by a factor of 2.5 for each unit increase in [tex]x[/tex].
D. The domain of the function is all real numbers.
E. The range of the function is all real numbers greater than 3.

Sagot :

GinnyAnswer
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].
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