1. **State the problem:** We have the function $f(x) = \log_{\frac{1}{9}} x$ and need to determine which of the given statements about $f(x)$ is true. 2. **Recall properties of logarithmic functions:** - The base $a$ of a logarithm $\log_a x$ must be positive and not equal to 1. - If $0 < a < 1$, the logarithmic function is **decreasing**. - The vertical asymptote of $\log_a x$ is at $x=0$. - The function is defined for $x > 0$. - The $y$-intercept occurs when $x=1$ because $\log_a 1 = 0$ for any valid base $a$. 3. **Analyze the base:** Here, the base is $\frac{1}{9}$, which is between 0 and 1, so $f(x)$ is a decreasing function. 4. **Check each statement:** - "$f$ is an increasing function": False, since base $\frac{1}{9} < 1$ implies decreasing. - "$y=0$ is asymptote of $f$": False, the vertical asymptote is at $x=0$, not $y=0$. - "For each $x \in (0,1)$, $f(x) > 0$": Since $f(x) = \log_{\frac{1}{9}} x$, and $x$ is between 0 and 1, for base less than 1, $\log_a x > 0$ when $x \in (0,1)$. This is true because logarithm with base less than 1 reverses the inequality. - "It has $y$ intercept": False, logarithmic functions do not have $y$-intercepts because they are undefined at $x=0$. 5. **Conclusion:** The true statement is: "For each $x \in (0,1)$, $f(x) > 0$." Final answer: **For each $x \in (0,1)$, $f(x) > 0$.**