What is the range of the function $y=2 e^x-1$?
A. all real numbers less than -1
B. all real numbers greater than -1
C. all real numbers less than 1
D. all real numbers greater than 1
Answer (2)
The exponential function e x is always positive, with a range of ( 0 , ∞ ) .
Multiplying by 2, the range of 2 e x remains ( 0 , ∞ ) .
Subtracting 1 shifts the range to ( − 1 , ∞ ) .
The range of y = 2 e x − 1 is all real numbers greater than -1, so the answer is all real numbers greater than -1 .
Explanation
Understanding the Function We are asked to find the range of the function y = 2 e x − 1 . To do this, we need to understand how the exponential function e x behaves.
Analyzing the Exponential Function The exponential function e x is always positive for any real number x . As x approaches − ∞ , e x approaches 0, but never actually reaches 0. As x approaches + ∞ , e x also approaches + ∞ . Therefore, the range of e x is ( 0 , ∞ ) .
Considering the Transformation Now, let's consider the function 2 e x . Since e x is always positive, 2 e x is also always positive. Multiplying by 2 simply stretches the function vertically, but it doesn't change the range. So, the range of 2 e x is still ( 0 , ∞ ) .
Determining the Final Range Finally, we have the function y = 2 e x − 1 . Subtracting 1 from 2 e x shifts the entire function down by 1 unit. Since the range of 2 e x is ( 0 , ∞ ) , subtracting 1 from every value in this range gives us ( − 1 , ∞ ) . This means that y can take any value greater than -1, but it can never be equal to or less than -1.
Final Answer Therefore, the range of the function y = 2 e x − 1 is all real numbers greater than -1.
Examples
Imagine you're tracking the growth of a bacteria population. The number of bacteria can be modeled by an exponential function. If the equation is y = 2 e x − 1 , where y is the number of bacteria and x is time, then knowing the range of this function tells you the possible values for the bacteria population. In this case, the population will always be greater than -1 (although negative bacteria don't make sense in the real world, mathematically, the function's range is still all numbers greater than -1). This kind of analysis is crucial in many scientific and engineering applications.
The range of the function y = 2 e x − 1 is all real numbers greater than -1, which can be expressed as the interval ( − 1 , + ∞ ) . Therefore, the correct answer is option B: all real numbers greater than -1.
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