Answer :
To tackle the problem of identifying the features of the functions [tex]\( f(x) = \log_4 x \)[/tex] and [tex]\( g(x) = \log_{0.25} x \)[/tex], we need to consider each function's properties such as domain, range, intercepts, behavior as [tex]\( x \)[/tex] increases, and asymptotes.
### Function Analysis
#### For [tex]\( f(x) = \log_4 x \)[/tex]:
1. Domain: The log function is defined for [tex]\( x > 0 \)[/tex]. Therefore, the domain of [tex]\( f(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex].
2. Range: The output of the log function can be any real number, so the range is [tex]\( (-\infty, \infty) \)[/tex].
3. Asymptote: There is a vertical asymptote at [tex]\( x = 0 \)[/tex].
4. x-intercept: [tex]\( f(x) = 0 \)[/tex] when [tex]\( \log_4 x = 0 \)[/tex]. This happens at [tex]\( x = 1 \)[/tex], so the x-intercept is [tex]\( (1, 0) \)[/tex].
5. Behavior:
- [tex]\( f(x) \)[/tex] is positive when [tex]\( x > 1 \)[/tex].
- [tex]\( f(x) \)[/tex] is negative when [tex]\( 0 < x < 1 \)[/tex].
- [tex]\( f(x) \)[/tex] is an increasing function as [tex]\( x \)[/tex] increases.
#### For [tex]\( g(x) = \log_{0.25} x \)[/tex]:
1. Domain: Like [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex] is defined only for [tex]\( x > 0 \)[/tex]. So the domain is [tex]\( (0, \infty) \)[/tex].
2. Range: The range is all real numbers [tex]\( (-\infty, \infty) \)[/tex].
3. Asymptote: There is a vertical asymptote at [tex]\( x = 0 \)[/tex].
4. x-intercept: [tex]\( g(x) = 0 \)[/tex] when [tex]\( \log_{0.25} x = 0 \)[/tex]. This happens at [tex]\( x = 1 \)[/tex], so the x-intercept is [tex]\( (1, 0) \)[/tex].
5. Behavior:
- [tex]\( g(x) \)[/tex] is positive when [tex]\( 0 < x < 1 \)[/tex].
- [tex]\( g(x) \)[/tex] is negative when [tex]\( x > 1 \)[/tex].
- [tex]\( g(x) \)[/tex] is a decreasing function as [tex]\( x \)[/tex] increases.
### Conclusion and Placement in Table
Given the analysis, we can summarize the features in the table:
#### Features of [tex]\( f \)[/tex] only:
- Positive over the interval [tex]\( (1, \infty) \)[/tex]
- Negative over the interval [tex]\( (0, 1) \)[/tex]
- Increasing as [tex]\( x \)[/tex] increases
#### Features of both [tex]\( f \)[/tex] and [tex]\( g \)[/tex]:
- Domain of [tex]\( (0, \infty) \)[/tex]
- Range of [tex]\( (-\infty, \infty) \)[/tex]
- Asymptote of [tex]\( x = 0 \)[/tex]
- [tex]\( x \)[/tex]-intercept of [tex]\( (1, 0) \)[/tex]
#### Features of [tex]\( g \)[/tex] only:
- Positive over the interval [tex]\( (0, 1) \)[/tex]
- Negative over the interval [tex]\( (1, \infty) \)[/tex]
- Decreasing as [tex]\( x \)[/tex] increases
Thus, the completed table will look like this:
\begin{tabular}{|l|l|l|}
\hline
Features of [tex]\( f \)[/tex] only & Features of both [tex]\( f \)[/tex] and [tex]\( g \)[/tex] & Features of [tex]\( g \)[/tex] only \\
\hline
Positive over the interval [tex]\( (1, \infty) \)[/tex] & Domain of [tex]\( (0, \infty) \)[/tex] & Positive over the interval [tex]\( (0, 1) \)[/tex] \\
Negative over the interval [tex]\( (0, 1) \)[/tex] & Range of [tex]\( (-\infty, \infty) \)[/tex] & Negative over the interval [tex]\( (1, \infty) \)[/tex] \\
Increasing as [tex]\( x \)[/tex] increases & Asymptote of [tex]\( x = 0 \)[/tex] & Decreasing as [tex]\( x \)[/tex] increases \\
& [tex]\( x \)[/tex]-intercept of [tex]\( (1, 0) \)[/tex] & \\
\hline
\end{tabular}
This table summarizes the distinctive and common features of [tex]\( f(x) = \log_4 x \)[/tex] and [tex]\( g(x) = \log_{0.25} x \)[/tex].
### Function Analysis
#### For [tex]\( f(x) = \log_4 x \)[/tex]:
1. Domain: The log function is defined for [tex]\( x > 0 \)[/tex]. Therefore, the domain of [tex]\( f(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex].
2. Range: The output of the log function can be any real number, so the range is [tex]\( (-\infty, \infty) \)[/tex].
3. Asymptote: There is a vertical asymptote at [tex]\( x = 0 \)[/tex].
4. x-intercept: [tex]\( f(x) = 0 \)[/tex] when [tex]\( \log_4 x = 0 \)[/tex]. This happens at [tex]\( x = 1 \)[/tex], so the x-intercept is [tex]\( (1, 0) \)[/tex].
5. Behavior:
- [tex]\( f(x) \)[/tex] is positive when [tex]\( x > 1 \)[/tex].
- [tex]\( f(x) \)[/tex] is negative when [tex]\( 0 < x < 1 \)[/tex].
- [tex]\( f(x) \)[/tex] is an increasing function as [tex]\( x \)[/tex] increases.
#### For [tex]\( g(x) = \log_{0.25} x \)[/tex]:
1. Domain: Like [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex] is defined only for [tex]\( x > 0 \)[/tex]. So the domain is [tex]\( (0, \infty) \)[/tex].
2. Range: The range is all real numbers [tex]\( (-\infty, \infty) \)[/tex].
3. Asymptote: There is a vertical asymptote at [tex]\( x = 0 \)[/tex].
4. x-intercept: [tex]\( g(x) = 0 \)[/tex] when [tex]\( \log_{0.25} x = 0 \)[/tex]. This happens at [tex]\( x = 1 \)[/tex], so the x-intercept is [tex]\( (1, 0) \)[/tex].
5. Behavior:
- [tex]\( g(x) \)[/tex] is positive when [tex]\( 0 < x < 1 \)[/tex].
- [tex]\( g(x) \)[/tex] is negative when [tex]\( x > 1 \)[/tex].
- [tex]\( g(x) \)[/tex] is a decreasing function as [tex]\( x \)[/tex] increases.
### Conclusion and Placement in Table
Given the analysis, we can summarize the features in the table:
#### Features of [tex]\( f \)[/tex] only:
- Positive over the interval [tex]\( (1, \infty) \)[/tex]
- Negative over the interval [tex]\( (0, 1) \)[/tex]
- Increasing as [tex]\( x \)[/tex] increases
#### Features of both [tex]\( f \)[/tex] and [tex]\( g \)[/tex]:
- Domain of [tex]\( (0, \infty) \)[/tex]
- Range of [tex]\( (-\infty, \infty) \)[/tex]
- Asymptote of [tex]\( x = 0 \)[/tex]
- [tex]\( x \)[/tex]-intercept of [tex]\( (1, 0) \)[/tex]
#### Features of [tex]\( g \)[/tex] only:
- Positive over the interval [tex]\( (0, 1) \)[/tex]
- Negative over the interval [tex]\( (1, \infty) \)[/tex]
- Decreasing as [tex]\( x \)[/tex] increases
Thus, the completed table will look like this:
\begin{tabular}{|l|l|l|}
\hline
Features of [tex]\( f \)[/tex] only & Features of both [tex]\( f \)[/tex] and [tex]\( g \)[/tex] & Features of [tex]\( g \)[/tex] only \\
\hline
Positive over the interval [tex]\( (1, \infty) \)[/tex] & Domain of [tex]\( (0, \infty) \)[/tex] & Positive over the interval [tex]\( (0, 1) \)[/tex] \\
Negative over the interval [tex]\( (0, 1) \)[/tex] & Range of [tex]\( (-\infty, \infty) \)[/tex] & Negative over the interval [tex]\( (1, \infty) \)[/tex] \\
Increasing as [tex]\( x \)[/tex] increases & Asymptote of [tex]\( x = 0 \)[/tex] & Decreasing as [tex]\( x \)[/tex] increases \\
& [tex]\( x \)[/tex]-intercept of [tex]\( (1, 0) \)[/tex] & \\
\hline
\end{tabular}
This table summarizes the distinctive and common features of [tex]\( f(x) = \log_4 x \)[/tex] and [tex]\( g(x) = \log_{0.25} x \)[/tex].
