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 increase as x increases.
The point ( 1 , 0 ) is not on the graph of y = b x when 1"> b > 1 .
Exponential functions y = b x with 1"> b > 1 approach 0 as x decreases towards negative infinity.
Therefore, properties I and IV are present, and the answer is I an d I V .
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 , as x increases, b x also increases. For example, if b = 2 , then as x goes from 1 to 2, y goes from 2 1 = 2 to 2 2 = 4 . So, this property is true.
Evaluating Property II Property II states: The point ( 1 , 0 ) exists in the table. This means when x = 1 , y = 0 . However, y = b 1 = b . Since 1"> b > 1 , y cannot be 0. So, this property is false.
Evaluating Property III Property III states: As the x -values increase, the y -values decrease. This is the opposite of property I, and since 1"> b > 1 , this property 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 goes to − ∞ , y = 2 x approaches 0. So, this property 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 has the same properties as described in the problem: as time (x) increases, the population (y) increases, and as time goes backwards to negative infinity, the population approaches zero.
The properties present for the exponential function y = b x where 1"> b > 1 are that as x increases, y increases (Property I), and as x decreases, y approaches 0 (Property IV). Properties II and III are false. Hence, the answer is I and I V .
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