Answer:
Parent function
[tex]f(x)=\log_2(x)[/tex]
Since we cannot take logs of zero or negative numbers, the domain is [tex](0, \infty)[/tex] and there is a vertical asymptote at [tex]x=0[/tex].
There are no horizontal asymptotes and the range is [tex](- \infty,\infty)[/tex].
The end behaviors of the parent function are:
[tex]\textsf{As } x \rightarrow 0^+, f(x) \rightarrow - \infty[/tex]
[tex]\textsf{As } x \rightarrow \infty, f(x) \rightarrow \infty[/tex]
To determine the attributes of the given function, compare the given function with the parent function.
Given function
[tex]f(x)=- \log_2(x)+5[/tex]
The given function is reflected in the x-axis. Therefore, the end behaviors are a reflection of those of the parent function:
[tex]\textsf{As } x \rightarrow 0^+, f(x) \rightarrow \infty[/tex]
[tex]\textsf{As } x \rightarrow \infty, f(x) \rightarrow - \infty[/tex]
As the given function is translated vertically (5 units up) only, the domain, range and asymptote will not change, since the function has not been translated horizontally.
[tex]\textsf{Domain}: \quad (0, \infty)[/tex]
[tex]\textsf{Range}: \quad (- \infty, \infty)[/tex]
[tex]\textsf{Asymptote}: \quad x=0[/tex]
Conclusion
The function f(x) is a [tex]\boxed{\sf logarithmic}}[/tex] function with a [tex]\boxed{\sf vertical}}[/tex] asymptote of [tex]\boxed{x = 0}[/tex].
The range of the function is [tex]\boxed{(- \infty, \infty)}[/tex], and it is [tex]\boxed{\sf decreasing}[/tex] on its domain of [tex]\boxed{(0, \infty)}[/tex].
The end behavior on the LEFT side is as [tex]\boxed{x \rightarrow 0^+}[/tex], [tex]\boxed{f(x) \rightarrow \infty}[/tex], and the end behavior of the RIGHT side is [tex]\boxed{x \rightarrow \infty}[/tex], [tex]\boxed{f(x) \rightarrow -\infty}[/tex].
