Which function has a range of [tex]y\ \textless \ 3[/tex]?
[tex]y=3(2)^x[/tex]
[tex]y=2(3)^x[/tex]
[tex]y=-(2)^x+3[/tex]
[tex]y=(2)^x-3[/tex]
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 . The functions are: y = 3 ( 2 ) x , y = 2 ( 3 ) x , y = − ( 2 ) x + 3 , and y = ( 2 ) x − 3 . We need to determine the range of each function and see which one satisfies the condition y < 3 .
Analyzing the Range of Each Function Let's analyze the range of each function:
y = 3 ( 2 ) x : Since 2 x is always positive for any real number x , 3 ( 2 ) x will also always be positive. As x approaches − ∞ , 2 x approaches 0, thus y approaches 0. As x approaches ∞ , 2 x approaches ∞ , thus y approaches ∞ . Therefore, the range of this function is 0"> y > 0 .
y = 2 ( 3 ) x : Similar to the previous function, 3 x is always positive for any real number x , so 2 ( 3 ) x will also always be positive. As x approaches − ∞ , 3 x approaches 0, thus y approaches 0. As x approaches ∞ , 3 x approaches ∞ , thus y approaches ∞ . Therefore, the range of this function is 0"> y > 0 .
y = − ( 2 ) x + 3 : Since 2 x is always positive, − ( 2 ) x will always be negative. As x approaches − ∞ , 2 x approaches 0, so − ( 2 ) x approaches 0, and y approaches 3 . As x approaches ∞ , 2 x approaches ∞ , so − ( 2 ) x approaches − ∞ , and y approaches − ∞ . Therefore, the range of this function is y < 3 .
y = ( 2 ) x − 3 : As x approaches − ∞ , 2 x approaches 0, so y approaches − 3 . As x approaches ∞ , 2 x approaches ∞ , so y approaches ∞ . Therefore, the range of this function is -3"> y > − 3 .
Finding the Function with the Desired Range Comparing the ranges of the four functions to the condition y < 3 , we find that only the function y = − ( 2 ) x + 3 has a range that satisfies this condition.
Final Answer Therefore, the function with a range of y < 3 is y = − ( 2 ) x + 3 .
Examples
Understanding the range of exponential functions is crucial in various real-world scenarios. For instance, consider the decay of a radioactive substance. The amount of the substance remaining over time can be modeled by an exponential function. If we know that the amount of substance must always be less than a certain value (e.g., due to safety regulations), we need to determine the parameters of the exponential decay function to ensure that this condition is always met. Similarly, in financial models, understanding the range of exponential growth or decay functions helps in predicting investment outcomes and managing risks.
