Which function has a range of $y<3$?

$y=3(2)^x$
$y=2(3)^x$
$y=-(2)^x+3$
$y=(2)^x-3$

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 , y = ( 2 ) x − 3 . We need to determine the range of each function and check if it satisfies the condition y < 3 .

Analyzing the Range of Each Function Let's analyze the range of each function:

y = 3 ( 2 ) x :

Since 0"> 2 x > 0 for all x , then 0"> 3 ( 2 ) x > 0 for all x .
As x approaches − ∞ , 3 ( 2 ) x approaches 0.
As x approaches ∞ , 3 ( 2 ) x approaches ∞ .
Therefore, the range is 0"> y > 0 .


y = 2 ( 3 ) x :

Since 0"> 3 x > 0 for all x , then 0"> 2 ( 3 ) x > 0 for all x .
As x approaches − ∞ , 2 ( 3 ) x approaches 0.
As x approaches ∞ , 2 ( 3 ) x approaches ∞ .
Therefore, the range is 0"> y > 0 .


y = − ( 2 ) x + 3 :

Since 0"> 2 x > 0 for all x , then − ( 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 − ∞ .
Therefore, the range is y < 3 .


y = ( 2 ) x − 3 :

Since 0"> 2 x > 0 for all x , then -3"> ( 2 ) x − 3 > − 3 .
As x approaches − ∞ , ( 2 ) x approaches 0, so y approaches − 3 .
As x approaches ∞ , ( 2 ) x approaches ∞ , so y approaches ∞ .
Therefore, the range is -3"> y > − 3 .


Identifying the Function with Range y<3 Comparing the ranges of the four functions with the condition y < 3 , we find that only the function y = − ( 2 ) x + 3 satisfies the condition.

Final Answer 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, in modeling the decay of radioactive substances, the amount of remaining substance decreases exponentially over time. Similarly, in financial models, understanding the range of exponential functions helps in predicting the depreciation of assets or the growth of investments under certain conditions. These models rely on the properties of exponential functions to provide accurate predictions within specific boundaries.

Answered by GinnyAnswer | 2025-07-07