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 , ∞ ) .
The domain of this function is ( − ∞ , ∞ ) The range of this function is ( 0 , ∞ )
Explanation
Understanding Domain and Range We are given the function y = 3 c d o t 5 x and we need to identify its domain and range. The domain represents all possible input values (x-values) for which the function is defined, and the range represents all possible output values (y-values) that the function can produce.
Determining the Domain The function y = 3 c d o t 5 x is an exponential function. Exponential functions are defined for all real numbers. This means that we can plug in any real number for x and get a valid output. Therefore, the domain of the function is all real numbers.
Determining the Range The exponential term 5 x is always positive for any real number x . As x approaches − ∞ , 5 x approaches 0, but never actually reaches 0. As x approaches ∞ , 5 x approaches ∞ . Since we are multiplying 5 x by 3, the result 3 c d o t 5 x will also always be positive. Therefore, the range of the function is all positive real numbers.
Expressing Domain and Range in Interval Notation In interval notation, the domain is ( − ∞ , ∞ ) and the range is ( 0 , ∞ ) .
Final Answer The domain of the function y = 3 c d o t 5 x is all real numbers, and the range is all positive real numbers.
Examples
Exponential functions are used to model various real-world phenomena, such as 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 in the account over time can be modeled by an exponential function. Understanding the domain and range of these functions helps you understand the possible values of the variables in the model and make predictions about the future.
