Question

which statements are true of the function ( f(x) = 3(2.5)^x )? choose three correct answers.
the domain of the function is all real numbers.
the function increases by a factor of 2.5 for each unit increase in ( x ).
the range of the function is all real numbers greater than 3.
the function is exponential.
the initial value of the function is 2.5.

Explanation:

Step1: Identify function type

The function $f(x)=3(2.5)^x$ follows the exponential form $f(x)=ab^x$ where $a
eq0$, $b>0,b
eq1$, so it is exponential.

Step2: Determine the domain

For exponential functions, $x$ can be any real number, so domain is all real numbers.

Step3: Check growth factor

For $f(x)=ab^x$, each unit increase in $x$ multiplies the output by $b$. Here $b=2.5$, so the function increases by a factor of 2.5 per $x$ unit.

Step4: Verify initial value

Initial value is $f(0)=3(2.5)^0=3$, not 2.5, so this is false.

Step5: Verify the range

$2.5^x>0$ for all real $x$, so $f(x)=3(2.5)^x>0$, not all real numbers greater than 3, so this is false.

Answer:

1. The domain of the function is all real numbers.
2. The function is exponential.
3. The function increases by a factor of 2.5 for each unit increase in $x$.