identifying characteristics of an exponential growth function

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.

Question

identifying characteristics of an exponential growth function

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.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Exponential Functions", "y-intercept" ], "new_concepts": [ "Exponential Growth", "Domain and Range" ], "current_concepts": [ "Exponential Functions", "y-intercept", "Exponential Growth", "Domain and Range" ] } ## Analyze the function type and initial value $$ \begin{aligned} &f(x) = 3(2.5)^x \ &\text{Type: Exponential function of form } f(x) = ab^x \ &\text{Initial value (y-intercept): } f(0) = 3(2.5)^0 = 3 eq 2.5 \end{aligned} $$ ## Determine growth factor and domain $$ \begin{aligned} &\text{Growth factor: } b = 2.5 \ &\frac{f(x+1)}{f(x)} = \frac{3(2.5)^{x+1}}{3(2.5)^x} = 2.5 \ &\text{Domain: } x \in (-\infty, \infty) \end{aligned} $$ ## Determine range $$ \begin{aligned} &\text{Since } 2.5^x > 0 \text{ for all } x \in \mathbb{R}: \ &f(x) = 3(2.5)^x > 0 \ &\text{Range: } y \in (0, \infty) eq (3, \infty) \end{aligned} $$

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.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Exponential Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the function type and initial value

Determine growth factor and domain

Determine range

Answer: