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 exponential function e x is always positive.
Multiplying by 4, the function 4 e x is also always positive.
As x approaches − ∞ , 4 e x approaches 0.
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 the function can produce.
Analyzing the Exponential Function The exponential function e x is defined for all real numbers x , and its value is always positive. That is, 0"> e x > 0 for all x ∈ R .
Determining the Sign of the Function Since 0"> e x > 0 for all real numbers x , then 0"> 4 e x > 0 for all real numbers x . This means that the function y = 4 e x will only produce positive values.
Behavior as x Approaches Negative Infinity As x approaches negative infinity ( x → − ∞ ), e x approaches 0, but it never actually reaches 0. Therefore, 4 e x also approaches 0 as x → − ∞ .
Behavior as x Approaches Positive Infinity As x approaches positive infinity ( x → ∞ ), e x also approaches infinity. Therefore, 4 e x also approaches infinity as x → ∞ .
Conclusion Since 4 e x is always positive and can take any value greater than 0, the range of the function y = 4 e x is all real numbers greater than 0.
Examples
Consider a scenario where you are modeling the growth of a bacteria population. The number of bacteria at time t can be represented by the function N ( t ) = 4 e t , where t is time in hours. Since the exponential function is always positive, this model tells us that the bacteria population will always be greater than 0 and will increase exponentially over time. Understanding the range of exponential functions helps in predicting and interpreting such growth patterns in various real-world scenarios.
