Choose all that apply. Which of the following happens if you increase the value of [tex]$b$[/tex] in any exponential function of the form [tex]$y=a b^x$[/tex] where [tex]$b\ \textgreater \ 0$[/tex]?
I. The domain changes.
II. The range changes.
III. The [tex]$y$[/tex] values increase more quickly.
Answer (2)
The domain of y = a b x does not change when b changes.
The range of y = a b x does not change when b changes.
The y values increase more quickly when b increases.
Therefore, the correct answer is III only, so III o n l y .
Explanation
Problem Analysis Let's analyze the given statements about the exponential function y = a b x where 0"> b > 0 . We need to determine what happens when we increase the value of b .
Domain Analysis I. The domain changes.
The domain of an exponential function y = a b x is all real numbers. Changing the value of b does not affect the domain. Therefore, this statement is false.
Range Analysis II. The range changes.
The range of an exponential function y = a b x depends on the value of a . If 0"> a > 0 , the range is 0"> y > 0 . If a < 0 , the range is y < 0 . Changing the value of b does not affect the range. Therefore, this statement is false.
Rate of Increase Analysis III. The y values increase more quickly.
Let's consider an example. Let a = 2 and x = 3 . If b = 1.5 , then y = 2 × ( 1.5 ) 3 = 2 × 3.375 = 6.75 . If we increase b to 2.5 , then y = 2 × ( 2.5 ) 3 = 2 × 15.625 = 31.25 . As we can see, increasing the value of b results in a larger y value for the same x . This means that the y values increase more quickly as x increases. Therefore, this statement is true.
Conclusion Therefore, the correct answer is that only statement III is true.
Examples
Exponential functions are used to model population growth. If y represents the population size, x represents time, a represents the initial population, and b represents the growth rate, then increasing b means the population grows faster over time. For example, if a population starts at 100 and grows at a rate of 1.05 (5%) per year, the population after 10 years would be 100 × ( 1.05 ) 10 ≈ 162.89 . If the growth rate increases to 1.10 (10%) per year, the population after 10 years would be 100 × ( 1.10 ) 10 ≈ 259.37 . This shows how increasing the base b leads to faster population growth.
When the value of b is increased in the function y = a b x , the domain and range remain unchanged, but the y values will increase more quickly. Therefore, the only correct statement is III: "The y values increase more quickly." The final answer is III only.
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