Consider [tex]f(x)=b^x[/tex]. Which statement(s) are true for [tex]0\ \textless \ b\ \textless \ 1[/tex]? Check all that apply.
The domain is all real numbers.
The domain is [tex]x\ \textgreater \ 0[/tex].
The range is all real numbers.
The range is [tex]y\ \textgreater \ 0[/tex].
The graph has [tex]x[/tex]-intercept 1.
The graph has a [tex]y[/tex]-intercept of 1.
The function is always increasing.
The function is always decreasing.
Answer (1)
The domain of f ( x ) = b x is all real numbers.
The range of f ( x ) = b x is 0"> y > 0 .
The graph has a y -intercept of 1.
The function is always decreasing.
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.
Explanation
Understanding the Problem We are given the function f ( x ) = b x where 0 < b < 1 . We need to determine which of the given statements are true about this function.
Determining the Domain The domain of an exponential function f ( x ) = b x is all real numbers, regardless of the value of b . This is because we can raise b to any real power.
Determining the Range The range of an exponential function f ( x ) = b x where 0 < b < 1 is 0"> y > 0 . As x approaches infinity, b x approaches 0, but never actually reaches 0. As x approaches negative infinity, b x approaches infinity. Thus, the range is all positive real numbers.
Determining the x-intercept To find the x -intercept, we set f ( x ) = 0 and solve for x . However, b x is always greater than 0 for any real number x , so there is no x -intercept.
Determining the y-intercept To find the y -intercept, we set x = 0 and evaluate f ( 0 ) . We have f ( 0 ) = b 0 = 1 . So the y -intercept is 1.
Determining if the function is increasing or decreasing To determine if the function is increasing or decreasing, we can consider the derivative of f ( x ) . Since f ( x ) = b x , f ′ ( x ) = b x ln ( b ) . Since 0 < b < 1 , ln ( b ) < 0 . Therefore, f ′ ( x ) < 0 for all x , which means the function is always decreasing.
Conclusion Based on the above analysis, 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 you invest money in a savings account that compounds interest continuously, the amount of money you have after a certain period of time can be modeled by an exponential function. Similarly, the decay of a radioactive substance can be modeled by an exponential function with a base between 0 and 1.
