What are the domain and range of the function $f(x)=3^x+5$?
A. domain: $(-\infty, \infty)$; range: $(0, \infty)$
B. domain: $(-\infty, \infty)$; range: $(5, \infty)$
C. domain: $(0, \infty)$; range: $(-\infty, \infty)$
D. domain: $(5, \infty)$; range: $(-\infty, \infty)$
Answer (1)
The domain of f ( x ) = 3 x + 5 is all real numbers since 3 x is defined for all real numbers: ( − ∞ , ∞ ) .
The range of 3 x is ( 0 , ∞ ) .
Adding 5 to 3 x shifts the range upwards by 5 units, resulting in a range of ( 5 , ∞ ) for f ( x ) = 3 x + 5 .
The domain is ( − ∞ , ∞ ) and the range is ( 5 , ∞ ) .
Explanation
Understanding the Function We are asked to find the domain and range of the function f ( x ) = 3 x + 5 . Let's break this down.
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 the exponential function 3 x , we can input any real number for x . There are no restrictions, such as division by zero or taking the square root of a negative number. Therefore, the domain of 3 x is all real numbers, which can be written as ( − ∞ , ∞ ) . Since adding a constant (5 in this case) does not change the domain, the domain of f ( x ) = 3 x + 5 is also ( − ∞ , ∞ ) .
Determining the Range The range of a function is the set of all possible output values (y-values) that the function can produce. For the exponential function 3 x , the output is always positive. As x approaches − ∞ , 3 x approaches 0, but never actually reaches 0. As x approaches ∞ , 3 x also approaches ∞ . Therefore, the range of 3 x is ( 0 , ∞ ) . Now, let's consider the function f ( x ) = 3 x + 5 . Since we are adding 5 to 3 x , the range will be shifted upwards by 5 units. So, the range of f ( x ) = 3 x + 5 is ( 0 + 5 , ∞ + 5 ) , which simplifies to ( 5 , ∞ ) .
Final Answer Therefore, the domain of the function f ( x ) = 3 x + 5 is ( − ∞ , ∞ ) , and the range is ( 5 , ∞ ) .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. Understanding the domain and range of these functions helps us to interpret the models correctly. For example, if we are modeling the population of a city using an exponential function, the domain would represent the time period for which the model is valid, and the range would represent the possible population sizes. Knowing the range helps us understand the minimum and maximum population values that the model predicts.
