Question

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

Answer 1

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}`.