The image contains questions about the characteristics of the graph of the function $f(x) = \frac{1}{2} \log x$. Let's analyze each statement given in the image to determine which are correct.

Analysis of the Function $f(x) = \frac{1}{2} \log x$:

1. Domain of $f(x)$: The domain of $f(x) = \frac{1}{2} \log x$ is all positive real numbers ($x > 0$), because the logarithmic function is defined only for positive values of $x$.

2. Range of $f(x)$: The range of $f(x)$ is all real numbers ($y \in \mathbb{R}$), because the logarithmic function can take any real value as $x$ increases or decreases within its domain.

3. Behavior near the axes:

   • The graph approaches the y-axis ($x = 0$) but does not touch or cross it, as logarithms are undefined at zero and negative values.
   • The graph does not intersect the x-axis, but $f(x)$ can approach negative infinity as $x$ approaches zero from the right.

4. Intersection with the axes:

   • It only intersects the x-axis at the point $(1, 0)$, as $\log(1) = 0$.
   • It never intersects the y-axis because the logarithmic function is not defined for $x \leq 0$.

5. Monotonicity: Since $\log x$ is an increasing function, $f(x) = \frac{1}{2} \log x$ is also increasing (monotonically).

Evaluating Statements:

1. Statement 1: Correct - Domain is all positive real numbers.
2. Statement 2: Correct - The range includes all real numbers.
3. Statement 3: Incorrect - The domain is not negative.
4. Statement 4: Correct - The graph approaches but does not intersect the y-axis.
5. Statement 5: Incorrect - The graph does not intersect the x-axis beyond $(1, 0)$.
6. Statement 6: Correct - Intersects at $(1, 0)$ and is monotonically increasing.
7. Statement 7: Incorrect - The function does not decrease.

Therefore, the correct statements are 1, 2, 4, and 6.

Would you like further details or explanations? Here are some related questions you might find interesting:

1. What is the transformation of a logarithmic function when multiplied by a constant?
2. How does the base of the logarithm affect the shape of the graph?
3. What happens to the graph of $f(x) = \frac{1}{2} \log x$ if the logarithm is taken to a different base?
4. How do you find the asymptotes of logarithmic functions?
5. What are the general properties of logarithmic functions?

Tip: To better understand logarithmic functions, always consider their domain, range, and asymptotic behavior.