What are the domain, range, and asymptote of [tex]h(x)=(0.5)^x-9[/tex]?
A. domain: [tex] \{x \mid x\ \textgreater \ 9\} [/tex]; range: [tex] \{y \mid y [/tex] is a real number [tex] \} [/tex]; asymptote: [tex]y=9[/tex]
B. domain: [tex] \{x \mid x\ \textgreater \ -9\} [/tex]; range: [tex] \{y \mid y [/tex] is a real number [tex] \} [/tex]; asymptote: [tex]y=-9[/tex]
C. domain: [tex] \{x \mid x [/tex] is a real number [tex] \} [/tex]; range: [tex] \{y \mid y\ \textgreater \ 9\} [/tex]; asymptote: [tex]y=9[/tex]
D. domain: [tex] \{x \mid x [/tex] is a real number [tex] \} [/tex]; range: [tex] \{y \mid y\ \textgreater \ -9\} [/tex]; asymptote: [tex]y=-9[/tex]
Answer (2)
The domain of h ( x ) = ( 0.5 ) x − 9 is all real numbers.
The range of h ( x ) is -9"> y > − 9 .
The horizontal asymptote of h ( x ) is y = − 9 .
Therefore, the correct answer is domain: {x \mid x is a real number } ; range: -9}"> y ∣ y > − 9 ; asymptote: y = − 9 . -9}; \text{ asymptote: } y=-9}}"> domain: x ∣ x is a real number ; range: y ∣ y > − 9 ; asymptote: y = − 9
Explanation
Analyzing the Function We are given the function h ( x ) = ( 0.5 ) x − 9 and need to determine its domain, range, and asymptote. Let's analyze each of these properties.
Determining the Domain The domain of a function is the set of all possible input values (x-values) for which the function is defined. For exponential functions of the form a x , where a is a positive number, the domain is all real numbers. Since h ( x ) is an exponential function with a vertical shift, its domain is also all real numbers.
Determining the Range The range of a function is the set of all possible output values (y-values). For the exponential function ( 0.5 ) x , the range is all positive real numbers, i.e., ( 0 , ∞ ) . The function h ( x ) = ( 0.5 ) x − 9 is a vertical shift of ( 0.5 ) x by -9 units. Therefore, the range of h ( x ) is all real numbers greater than -9, i.e., ( − 9 , ∞ ) . This can be written as -9"> y > − 9 .
Determining the Asymptote A horizontal asymptote is a horizontal line that the graph of the function approaches as x tends to positive or negative infinity. For the exponential function ( 0.5 ) x , the horizontal asymptote is y = 0 . Since h ( x ) = ( 0.5 ) x − 9 is a vertical shift of ( 0.5 ) x by -9 units, the horizontal asymptote of h ( x ) is y = − 9 .
Conclusion Therefore, the domain of h ( x ) is all real numbers, the range is -9"> y > − 9 , and the horizontal asymptote is y = − 9 .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. Understanding the domain, range, and asymptotes of exponential functions helps us to analyze and predict the behavior of these phenomena. For example, if we are modeling the decay of a radioactive substance with the function A ( t ) = A 0 ( 0.5 ) t − 5 , where A ( t ) is the amount of the substance remaining after time t , A 0 is the initial amount, and the -5 represents a constant background radiation level, then knowing the range and asymptote of this function helps us understand the minimum amount of the substance that will remain over time.
The function h ( x ) = ( 0.5 ) x − 9 has a domain of all real numbers, a range of -9"> y > − 9 , and a horizontal asymptote at y = − 9 . Thus, the correct choice is option D.
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