To solve the question of determining the domain, range, and asymptote of the function [tex]\( h(x) = (1.4)^x + 5 \)[/tex], let's analyze each of these components step-by-step:
1. Domain:
The function [tex]\( h(x) = (1.4)^x + 5 \)[/tex] involves an exponential term [tex]\((1.4)^x\)[/tex]. Exponential functions are defined for all real numbers. This means that you can substitute any real number [tex]\( x \)[/tex] into [tex]\((1.4)^x\)[/tex] and get a valid output. Therefore, the domain of [tex]\( h(x) \)[/tex] is:
[tex]\[
\{ x \mid x \text{ is a real number} \}
\][/tex]
2. Range:
Next, let's consider the range of the function. The exponential function [tex]\((1.4)^x\)[/tex] is always positive, and it grows as [tex]\( x \)[/tex] increases. Specifically:
- When [tex]\( x \to -\infty \)[/tex], [tex]\((1.4)^x \to 0 \)[/tex].
- When [tex]\( x \to \infty \)[/tex], [tex]\((1.4)^x \to \infty \)[/tex].
Since we add 5 to the exponential function, the smallest value [tex]\( h(x) \)[/tex] can approach is:
[tex]\[
0 + 5 = 5
\][/tex]
but it will never actually reach 5; it will just get arbitrarily close to 5 from above. Therefore, [tex]\( h(x) \)[/tex] is always greater than 5, and increases infinitely as [tex]\( x \)[/tex] increases. Hence, the range of [tex]\( h(x) \)[/tex] is:
[tex]\[
\{ y \mid y > 5 \}
\][/tex]
3. Asymptote:
An asymptote is a line that a graph approaches but never touches. For this function, as [tex]\( x \to -\infty \)[/tex], [tex]\((1.4)^x \)[/tex] approaches 0. Thus:
[tex]\[
h(x) = (1.4)^x + 5 \to 0 + 5 = 5
\][/tex]
So, the function approaches but never actually reaches [tex]\( y = 5 \)[/tex]. Therefore, the horizontal asymptote is:
[tex]\[
y = 5
\][/tex]
Summarizing the above analysis, the correct answers are:
- Domain: [tex]\(\{ x \mid x \text{ is a real number} \}\)[/tex]
- Range: [tex]\(\{ y \mid y > 5 \}\)[/tex]
- Asymptote: [tex]\( y = 5 \)[/tex]
Thus, the correct choice is:
domain: [tex]\( \{x \mid x \text{ is a real number}\} \)[/tex]; range: [tex]\( \{y \mid y > 5 \}\)[/tex]; asymptote: [tex]\( y = 5 \)[/tex]
