Which function has a range of $y<3$?
A. $y=3(2)^x$
B. $y=2(3)^x$
C. $y=-(2)^x+3$
D. $y=(2)^x-3$
Answer (1)
Analyze the range of y = 3 ( 2 ) x , which is 0"> y > 0 .
Analyze the range of y = 2 ( 3 ) x , which is 0"> y > 0 .
Analyze the range of y = − ( 2 ) x + 3 , which is y < 3 .
Analyze the range of y = ( 2 ) x − 3 , which is -3"> y > − 3 .
The function with a range of y < 3 is y = − ( 2 ) x + 3 .
Explanation
Understanding the Problem We are given four functions and asked to identify the one with a range of y < 3 .
Listing the Functions The functions are: y = 3 ( 2 ) x y = 2 ( 3 ) x y = − ( 2 ) x + 3 y = ( 2 ) x − 3
Objective We need to determine the range of each function and see which one satisfies the condition y < 3 .
Analyzing the first function Let's analyze the range of the function y = 3 ( 2 ) x . Since 0"> 2 x > 0 for all x , 0"> 3 ( 2 ) x > 0 for all x . Also, as x approaches − ∞ , 3 ( 2 ) x approaches 0, and as x approaches ∞ , 3 ( 2 ) x approaches ∞ . Thus, the range is 0"> y > 0 .
Analyzing the second function Now, let's analyze the range of the function y = 2 ( 3 ) x . Since 0"> 3 x > 0 for all x , 0"> 2 ( 3 ) x > 0 for all x . Also, as x approaches − ∞ , 2 ( 3 ) x approaches 0, and as x approaches ∞ , 2 ( 3 ) x approaches ∞ . Thus, the range is 0"> y > 0 .
Analyzing the third function Next, let's analyze the range of the function y = − ( 2 ) x + 3 . Since 0"> 2 x > 0 for all x , − ( 2 ) x < 0 for all x . As x approaches − ∞ , − ( 2 ) x approaches 0, so y approaches 3. As x approaches ∞ , − ( 2 ) x approaches − ∞ , so y approaches − ∞ . Thus, the range is y < 3 .
Analyzing the fourth function Finally, let's analyze the range of the function y = ( 2 ) x − 3 . Since 0"> 2 x > 0 for all x , -3"> ( 2 ) x − 3 > − 3 for all x . As x approaches − ∞ , ( 2 ) x approaches 0, so y approaches − 3 . As x approaches ∞ , ( 2 ) x approaches ∞ , so y approaches ∞ . Thus, the range is -3"> y > − 3 .
Conclusion Comparing the ranges of the four functions to the condition y < 3 , we find that the function y = − ( 2 ) x + 3 has a range of y < 3 . Therefore, it is the correct answer.
Examples
Understanding the range of functions is crucial in many real-world applications. For example, when modeling population growth or radioactive decay, exponential functions are often used. Knowing the range helps us determine the possible values of the population or the amount of radioactive material remaining over time. In finance, understanding the range of investment returns can help assess the potential risks and rewards. This concept is fundamental in various fields, providing a framework for predicting and interpreting real-world phenomena.
