Identify the domain and range of each function.
$y=3 \cdot 5^x$
The domain of this function is
$\square$
The range of this function is
$\square$
Answer (1)
The domain of the exponential function y = 3 c d o t 5 x is all real numbers.
The range of the exponential function y = 3 c d o t 5 x is all positive real numbers.
The domain in interval notation is ( − ∞ , ∞ ) .
The range in interval notation is ( 0 , ∞ ) .
( − ∞ , ∞ ) ( 0 , ∞ )
Explanation
Analyzing the Function Let's analyze the given function to determine its domain and range. The function is an exponential function of the form y = 3 c d o t 5 x . We need to identify all possible values of x (the domain) and all possible values of y (the range).
Determining the Domain The domain of a function is the set of all possible input values (x-values) for which the function is defined. For exponential functions of the form a x (where 0"> a > 0 ), the domain is all real numbers. Multiplying by a constant (in this case, 3) does not change the domain. Therefore, the domain of y = 3 c d o t 5 x is all real numbers.
Determining the Range The range of a function is the set of all possible output values (y-values) that the function can produce. For an exponential function 5 x , the range is all positive real numbers (i.e., 0"> 5 x > 0 for all x ). Since we have y = 3 c d o t 5 x , we are multiplying the exponential function by 3. This means that the output will always be positive, and the range will be all positive real numbers.
Stating the Domain and Range In interval notation, the domain (all real numbers) is written as ( − ∞ , ∞ ) , and the range (all positive real numbers) is written as ( 0 , ∞ ) . Therefore, the domain of the function y = 3 c d o t 5 x is ( − ∞ , ∞ ) , and the range is ( 0 , ∞ ) .
Examples
Exponential functions are used in various real-world scenarios, such as modeling population growth, radioactive decay, and compound interest. For example, if you invest money in an account that earns compound interest, the amount of money you have over time can be modeled by an exponential function. Understanding the domain and range of these functions helps you predict the possible values and limitations of these models.
