What are the domain, range, and asymptote of [tex]h(x)=2^{x+4}[/tex]?
A. domain: [tex] \{x \mid x\ \textgreater \ 0\} [/tex]; range: [tex] \{y \mid y [/tex] is a real number [tex] \} [/tex]; asymptote: [tex]y=0[/tex]
B. domain: [tex] \{x \mid x\ \textgreater \ -4\} [/tex]; range: [tex] \{y \mid y [/tex] is a real number [tex] \} [/tex]; asymptote: [tex]y=-4[/tex]
C. domain: [tex] \{x \mid x [/tex] is a real number [tex] \} [/tex]; range: [tex] \{y \mid y\ \textgreater \ 0\} [/tex]; asymptote: [tex]y=0[/tex]
D. domain: [tex] \{x \mid x [/tex] is a real number [tex] \} [/tex]; range: [tex] \{y \mid y \geqslant 0\} [/tex]; asymptote: [tex]y=-4[/tex]
Answer (2)
The domain of h ( x ) = 2 x + 4 is all real numbers.
The range of h ( x ) = 2 x + 4 is all positive real numbers.
The horizontal asymptote of h ( x ) = 2 x + 4 is y = 0 .
Therefore, the domain is {x \mid x is a real number } , the range is 0}"> y ∣ y > 0 , and the asymptote is y = 0 . 0}; \text{ asymptote: } y=0}}"> domain: x ∣ x is a real number ; range: y ∣ y > 0 ; asymptote: y = 0
Explanation
Understanding the Problem The problem asks us to identify the domain, range, and asymptote of the exponential function h ( x ) = 2 x + 4 . Let's break down each of these concepts.
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 like h ( x ) = 2 x + 4 , there are no restrictions on the input. You can plug in any real number for x , and the function will produce a valid output. Therefore, the domain is all real numbers.
Determining the Range The range of a function is the set of all possible output values (y-values) that the function can produce. Exponential functions of the form a x (where 0"> a > 0 and a e q 1 ) always produce positive values. In our case, h ( x ) = 2 x + 4 will always be greater than 0, no matter what value we plug in for x . The function never actually reaches 0. Therefore, the range is all positive real numbers.
Determining the Asymptote An asymptote is a line that a curve approaches but never touches. For exponential functions of the form a x , there is a horizontal asymptote at y = 0 . This is because as x approaches negative infinity, the function gets closer and closer to 0, but never actually reaches it. The same is true for h ( x ) = 2 x + 4 . As x becomes very large in the negative direction, 2 x + 4 approaches 0. Therefore, the horizontal asymptote is y = 0 .
Final Answer In summary:
Domain: All real numbers
Range: All positive real numbers
Asymptote: y = 0
Therefore, the correct answer is: domain: {x \mid x is a real number } ; range: 0}"> y ∣ y > 0 ; asymptote: y = 0
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in an account that earns compound interest, the amount of money you have over time can be modeled by an exponential function. Understanding the domain, range, and asymptotes of these functions can help you make informed decisions about your investments.
The domain of h ( x ) = 2 x + 4 is all real numbers, the range is all positive real numbers, and the horizontal asymptote is y = 0 .
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