What is the range of the function $y=e^{4 x}$?
A. $y<0$
B. $y>0$
C. $y<4$
D. $y>4
Answer (1)
The function is an exponential function y = e 4 x .
Exponential functions e u are always positive, i.e., 0"> e u > 0 for all u .
Since 4 x can be any real number, e 4 x is always positive.
Therefore, the range of the function is 0}"> y > 0 .
Explanation
Understanding the Problem We are asked to find the range of the function y = e 4 x . This means we need to determine all possible values of y that the function can take.
Key Property of Exponential Functions The exponential function e u is always positive for any real number u . In other words, 0"> e u > 0 for all u .
Applying to the Given Function In our case, the exponent is 4 x . Since x can be any real number, 4 x can also be any real number. Therefore, we can say that 0"> y = e 4 x > 0 for all real values of x .
Determining the Range This means the function y = e 4 x can only take positive values. Therefore, the range of the function is 0"> y > 0 .
Final Answer The range of the function y = e 4 x is all positive real numbers, which can be written as 0"> y > 0 .
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 interest compounded continuously, the amount of money you have after a certain time can be modeled by an exponential function. Understanding the range of exponential functions helps you predict the possible outcomes in these scenarios.
