What is the range of the function $y=4 e^x$?
A. all real numbers greater than 0
B. all real numbers less than 0
C. all real numbers less than 4
D. all real numbers greater than 4
Answer (1)
The function is y = 4 e x .
The exponential function e x is always positive.
Therefore, 4 e x is always positive.
The range of y = 4 e x is all real numbers greater than 0: all real numbers greater than 0
Explanation
Understanding the Problem We are asked to find the range of the function y = 4 e x . The range of a function is the set of all possible output values (y-values) that we can get from the function.
Analyzing the Exponential Function The exponential function e x is defined for all real numbers x , and its output is always positive. That is, 0"> e x > 0 for all x ∈ R . As x approaches − ∞ , e x approaches 0, but never actually reaches 0. As x approaches + ∞ , e x approaches + ∞ .
Determining the Range Now, consider the function y = 4 e x . Since 0"> e x > 0 for all x , we have 0"> 4 e x > 0 for all x . This means that y is always positive. As x approaches − ∞ , 4 e x approaches 0, but never reaches 0. As x approaches + ∞ , 4 e x approaches + ∞ .
Final Answer Therefore, the range of the function y = 4 e x is all real numbers greater than 0.
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population starts with 4 individuals and grows exponentially with a growth rate of 100% per time unit, the population size at time t can be modeled by the function P ( t ) = 4 e t . The range of this function tells us all the possible population sizes that can occur, which in this case are all positive real numbers greater than 0.
