Question

What would i put in, in the blanks? please help, i don’t get this

Answer 1

Answer:

Parent function

$f(x)=\log_2(x)$

Since we cannot take logs of zero or negative numbers, the domain is $(0, \infty)$ and there is a vertical asymptote at $x=0$.

There are no horizontal asymptotes and the range is $(- \infty,\infty)$.

The end behaviors of the parent function are:

$\textsf{As } x \rightarrow 0^+, f(x) \rightarrow - \infty$

$\textsf{As } x \rightarrow \infty, f(x) \rightarrow \infty$

To determine the attributes of the given function, compare the given function with the parent function.

Given function

$f(x)=- \log_2(x)+5$

The given function is reflected in the x-axis.  Therefore, the end behaviors are a reflection of those of the parent function:

$\textsf{As } x \rightarrow 0^+, f(x) \rightarrow \infty$

$\textsf{As } x \rightarrow \infty, f(x) \rightarrow - \infty$

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.

$\textsf{Domain}: \quad (0, \infty)$

$\textsf{Range}: \quad (- \infty, \infty)$

$\textsf{Asymptote}: \quad x=0$

Conclusion

The function f(x) is a $\boxed{\sf logarithmic}}$ function with a $\boxed{\sf vertical}}$ asymptote of $\boxed{x = 0}$.  

The range of the function is $\boxed{(- \infty, \infty)}$, and it is $\boxed{\sf decreasing}$ on its domain of $\boxed{(0, \infty)}$.  

The end behavior on the LEFT side is as $\boxed{x \rightarrow 0^+}$, $\boxed{f(x) \rightarrow \infty}$, and the end behavior of the RIGHT side is $\boxed{x \rightarrow \infty}$, $\boxed{f(x) \rightarrow -\infty}$.