Question

which statements are true of the function ( f(x) = 3(2.5)^x )? check all that apply.

• the function is exponential.
• the initial value of the function is 2.5.
• the function increases by a factor of 2.5 for each unit increase in ( x ).
• the domain of the function is all real numbers.
• the range of the function is all real numbers greater than 3.

Explanation:

Step1: Identify function type

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

Step2: Find initial value

Initial value is $f(0)$. Calculate: $f(0)=3(2.5)^0=3\times1=3$.

Step3: Analyze growth factor

The base $b=2.5$, so the function multiplies by 2.5 per $x$ increase.

Step4: Determine domain

Exponential functions accept all real $x$, so domain is $\mathbb{R}$.

Step5: Determine range

Since $(2.5)^x>0$ for all $x$, $3(2.5)^x>0$. Range is all positive reals.

Answer:

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