To determine the domain, range, and asymptote of the function [tex]\( h(x) = 2^{x+4} \)[/tex], let us go through each part step by step.
1. Domain:
- The domain of an exponential function of the form [tex]\( a^{x+b} \)[/tex] where [tex]\( a > 0 \)[/tex] is all real numbers. This is because exponentiation is defined for all real numbers.
- Therefore, the domain of [tex]\( h(x) = 2^{x+4} \)[/tex] is [tex]\( \{ x \mid x \text{ is a real number} \} \)[/tex].
2. Range:
- The range of an exponential function [tex]\( a^{x+b} \)[/tex] where [tex]\( a > 0 \)[/tex] and [tex]\( a \neq 1 \)[/tex] is all positive real numbers. This is because an exponential function always grows and never touches zero or goes negative.
- Therefore, the range of [tex]\( h(x) = 2^{x+4} \)[/tex] is [tex]\( \{ y \mid y > 0 \} \)[/tex].
3. Asymptote:
- The horizontal asymptote of an exponential function [tex]\( a^{x+b} \)[/tex] is the horizontal line that the graph of the function approaches but never touches.
- For the function [tex]\( h(x) = 2^{x+4} \)[/tex], as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 2^{x+4} \)[/tex] approaches zero.
- Therefore, the horizontal asymptote of [tex]\( h(x) = 2^{x+4} \)[/tex] is [tex]\( y = 0 \)[/tex].
So, the correct answers are:
- Domain: [tex]\( \{ x \mid x \text{ is a real number} \} \)[/tex]
- Range: [tex]\( \{ y \mid y > 0 \} \)[/tex]
- Asymptote: [tex]\( y = 0 \)[/tex]
Hence, the correct option is:
Domain: [tex]\( \{ x \mid x \text{ is a real number} \} \)[/tex]; Range: [tex]\( \{ y \mid y > 0 \} \)[/tex]; Asymptote: [tex]\( y = 0 \)[/tex]
