Which properties are present in a table that represents an exponential function in the form [tex]$y=b^x$[/tex] when [tex]$b\ \textgreater \ 1$[/tex]?
I. As the [tex]$x$[/tex]-values increase, the [tex]$y$[/tex]-values increase.
II. The point [tex]$(1,0)$[/tex] exists in the table.
III. As the [tex]$x$[/tex]-values increase, the [tex]$y$[/tex]-values decrease.
IV. As the [tex]$x$[/tex]-values decrease, the [tex]$y$[/tex]-values decrease, approaching a singular value.
A. I and IV
B. I and II
C. II and III
D. III only
Answer (2)
Exponential functions y = b x with 1"> b > 1 exhibit specific properties.
As x increases, y increases, indicating exponential growth.
The function does not pass through the point ( 1 , 0 ) .
As x decreases, y decreases, approaching 0.
Therefore, the correct answer is I and IV: I and IV
Explanation
Analyzing the Problem We are given an exponential function in the form y = b x where 1"> b > 1 . We need to determine which of the given properties hold true for this function. Let's analyze each property individually.
Evaluating Property I Property I states: As the x -values increase, the y -values increase. Since 1"> b > 1 , the function is increasing. For example, if b = 2 , as x increases from 1 to 2, y increases from 2 1 = 2 to 2 2 = 4 . Thus, property I is true.
Evaluating Property II Property II states: The point ( 1 , 0 ) exists in the table. If x = 1 , then y = b 1 = b . Since 1"> b > 1 , y cannot be 0. Therefore, the point ( 1 , 0 ) does not exist in the table, and property II is false.
Evaluating Property III Property III states: As the x -values increase, the y -values decrease. Since 1"> b > 1 , as x increases, y increases, not decreases. Thus, property III is false.
Evaluating Property IV Property IV states: As the x -values decrease, the y -values decrease, approaching a singular value. As x approaches − ∞ , y = b x approaches 0. For example, if b = 2 , as x becomes very negative (e.g., -100), y = 2 − 100 which is very close to 0. So, property IV is true.
Conclusion Therefore, properties I and IV are true.
Examples
Exponential functions are used to model population growth. If a population starts at a size of 100 and grows by 5% each year, the population after x years can be modeled by the function y = 100 ( 1.05 ) x . This function exhibits the properties we discussed: as time (x) increases, the population (y) increases, and as time goes further back into the past (x decreases), the population approaches zero.
The properties of an exponential function y = b x with 1"> b > 1 show that as x increases, y also increases (I), and as x decreases, y approaches 0 (IV). Thus, the correct answer is I and IV.
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