Consider [tex]$f(x)=b^x$[/tex]. Which statement(s) are true for [tex]$0 < b < 1$[/tex]? Check all that apply.
A. The domain is all real numbers.
B. The domain is [tex]$x > 0$[/tex].
C. The range is all real numbers.
D. The range is [tex]$y > 0$[/tex].
E. The graph has [tex]$x$[/tex]-intercept 1.
F. The graph has a [tex]$y$[/tex]-intercept of 1.
G. The function is always increasing.
H. The function is always decreasing.
Answer (2)
The domain of f ( x ) = b x is all real numbers.
The range of f ( x ) = b x is 0"> y > 0 .
The graph of f ( x ) = b x has a y -intercept of 1.
The function f ( x ) = b x is always decreasing when 0 < b < 1 .
0, \text{ The graph has a y-intercept of 1, The function is always decreasing.}}"> The domain is all real numbers, The range is y > 0 , The graph has a y-intercept of 1, The function is always decreasing.
Explanation
Analyzing the Problem We are given the function f ( x ) = b x with the condition 0 < b < 1 . We need to determine which of the given statements are true about this function. Let's analyze each statement.
Determining the Domain The domain of an exponential function f ( x ) = b x is all real numbers, regardless of the value of b (as long as 0"> b > 0 ). So, the statement 'The domain is all real numbers' is true.
Determining the Range The range of an exponential function f ( x ) = b x is 0"> y > 0 , regardless of the value of b (as long as 0"> b > 0 and b e q 1 ). So, the statement 'The range is 0"> y > 0 ' is true, and the statement 'The range is all real numbers' is false.
Determining the x-intercept An x -intercept occurs when f ( x ) = 0 . For f ( x ) = b x , there is no value of x for which b x = 0 when 0 < b < 1 . Therefore, there is no x -intercept. The statement 'The graph has x -intercept 1' is false.
Determining the y-intercept A y -intercept occurs when x = 0 . For f ( x ) = b x , when x = 0 , f ( 0 ) = b 0 = 1 . So, the y -intercept is 1. The statement 'The graph has a y -intercept of 1' is true.
Determining if the function is increasing or decreasing When 0 < b < 1 , the exponential function f ( x ) = b x is a decreasing function. As x increases, b x decreases. For example, if b = 2 1 , then as x goes from 1 to 2, f ( x ) goes from 2 1 to 4 1 . Therefore, the statement 'The function is always decreasing' is true, and the statement 'The function is always increasing' is false.
Final Answer In summary, the true statements are:
The domain is all real numbers.
The range is 0"> y > 0 .
The graph has a y -intercept of 1.
The function is always decreasing.
Examples
Exponential functions are used to model various real-world phenomena, such as radioactive decay, population growth, and compound interest. For example, if a substance decays at a rate such that its mass is halved every year, the mass of the substance after x years can be modeled by f ( x ) = m 0 ⋅ ( 2 1 ) x , where m 0 is the initial mass. This is an example of an exponential function with 0 < b < 1 , similar to the one in the problem.
The true statements about the function f ( x ) = b x where 0 < b < 1 are that its domain is all real numbers, its range is 0"> y > 0 , it has a y-intercept of 1, and it is always decreasing.
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