Question

which statements are true about the graph of function f?
f(x) = \log x

\square \quad the graph has a domain of \\{x|0 < x < \infty\\} and approaches 0 as x decreases.
\square \quad the graph has a range of \\{y| -\infty < y < \infty\\} and decreases as x approaches 0.
\square \quad the graph has a domain of \\{x| -\infty < x < \infty\\} and approaches 0 as x decreases.
\square \quad the graph has a range of \\{y|0 < y < \infty\\} and decreases as x approaches 0.

Explanation:

Step1: Recall properties of \( f(x)=\log x \)

The function \( f(x)=\log x \) (assuming base 10 or natural log, properties hold) has:
- Domain: \( x > 0 \) (since logarithm is defined for positive real numbers), so domain is \( \{x|0 < x < \infty\} \).
- Range: All real numbers, \( \{y|-\infty < y < \infty\} \) (logarithm can take any real value).
- As \( x \) decreases (approaches 0 from the right), \( \log x \) approaches \( -\infty \) (so the function decreases as \( x \) approaches 0).
- As \( x \) increases, \( \log x \) increases, but for the statements:

Step2: Analyze each option

1. First option: "The graph has a domain of \( \{x|0 < x < \infty\} \) and approaches 0 as \( x \) decreases."
Wait, as \( x \) decreases (towards 0), \( \log x \) approaches \( -\infty \), not 0. Wait, maybe a typo? Wait, no—wait, \( \log 1 = 0 \). Wait, when \( x \) decreases from values greater than 1 towards 1, \( \log x \) decreases towards 0? Wait, no: \( \log x \) is increasing. So when \( x \) decreases (gets smaller), \( \log x \) decreases. So if \( x \) decreases towards 1, \( \log x \) decreases towards 0. But if \( x \) decreases towards 0, \( \log x \) decreases towards \( -\infty \). Wait, maybe the first option's "approaches 0" is when \( x \) decreases towards 1? No, the domain is \( x > 0 \). Wait, maybe I misread. Wait, let's re - check the second option: "The graph has a range of \( \{y|-\infty < y < \infty\} \) and decreases as \( x \) approaches 0."
The range is all real numbers, and as \( x \to 0^+ \), \( \log x \to -\infty \), so the function is decreasing as \( x \) approaches 0 (since as \( x \) gets smaller, \( f(x) \) gets smaller (more negative)).
The first option: domain is correct (\( x > 0 \)), but "approaches 0 as \( x \) decreases"—if \( x \) decreases from, say, 2 to 1, \( \log x \) goes from \( \log 2 \) to 0. If \( x \) decreases from 1 to 0.5, \( \log x \) goes from 0 to \( \log 0.5 \) (negative). So maybe the first option is partially correct? Wait, no, let's re - evaluate:

Wait, the standard properties:
- Domain of \( \log x \): \( (0, \infty) \) (so \( \{x|0 < x < \infty\} \))
- Range: \( (-\infty, \infty) \) (so \( \{y|-\infty < y < \infty\} \))
- As \( x \to 0^+ \), \( \log x \to -\infty \) (so the function is decreasing as \( x \) approaches 0 from the right)
- As \( x \to \infty \), \( \log x \to \infty \)

Now, let's check each option:

- Option 1: "The graph has a domain of \( \{x|0 < x < \infty\} \) and approaches 0 as \( x \) decreases."
When \( x \) decreases (towards 0), \( \log x \) approaches \( -\infty \), not 0. But if \( x \) decreases towards 1, \( \log x \) approaches 0. But the domain is \( x > 0 \), so maybe the wording is a bit off, but let's check other options.

- Option 2: "The graph has a range of \( \{y|-\infty < y < \infty\} \) and decreases as \( x \) approaches 0."
Range is correct (all real numbers), and as \( x \) approaches 0 (from the right), \( \log x \) decreases (goes to \( -\infty \)), so this is correct.

- Option 3: "The graph has a domain of \( \{x|-\infty < x < \infty\} \) and approaches 0 as \( x \) decreases."
Domain is wrong (logarithm is not defined for \( x \leq 0 \)), so this is incorrect.

- Option 4: "The graph has a range of \( \{y|0 < y < \infty\} \) and decreases as \( x \) approaches 0."
Range is wrong (range is all real numbers, not \( y > 0 \)), so this is incorrect.

Wait, but earlier analysis of option 1: domain is correct, but "approaches 0 as \( x \) decreases"—if \( x \) decreases from values greater than 1 to 1, \( \log x \) approaches 0. But if \( x \) decreases from 1 to 0, \( \log x \) goes negative. Maybe the first option's "approaches 0" is when \( x \) decreases towards 1? But the domain is \( x > 0 \). However, the second option: range is correct, and as \( x \) approaches 0, the function decreases (since \( \log x \) goes to \( -\infty \), so as \( x \) gets smaller, \( f(x) \) gets smaller). So the second option is correct. Also, let's re - check the first option: "approaches 0 as \( x \) decreases"—if \( x \) decreases (i.e., gets smaller), when \( x \) approaches 1 from the right (decreasing towards 1), \( \log x \) approaches 0. But the domain is \( x > 0 \), so maybe the first option is also correct? Wait, no, when \( x \) decreases (say, from 2 to 0.5), \( \log x \) goes from \( \log 2 \) to \( \log 0.5 \) (negative), so it doesn't approach 0 as \( x \) decreases in general. But the second option: range is correct, and as \( x \) approaches 0, the function decreases (since as \( x \) gets closer to 0, \( f(x) \) gets more negative, so it's decreasing). So the second option is correct. Also, let's check the first option again: domain is \( x > 0 \) (correct), and "approaches 0 as \( x \) decreases"—maybe when \( x \) decreases towards 1, but the statement says "as \( x \) decreases" (in general, as \( x \) gets smaller). So the first option is incorrect. The second option: range is all real numbers (correct), and as \( x \) approaches 0, the function decreases (since \( \log x \) is decreasing on \( (0, \infty) \), so as \( x \) approaches 0, \( f(x) \) decreases towards \( -\infty \)). So the second option is correct. Also, wait, maybe the first option has a mistake, but let's confirm:

The function \( y = \log x \):

- Domain: \( x \in (0, \infty) \) (so \( \{x|0 < x < \infty\} \))
- Range: \( y \in (-\infty, \infty) \) (so \( \{y|-\infty < y < \infty\} \))
- It is an increasing function? Wait, no! Wait, \( y=\log x \) (base > 1) is an increasing function. Wait, that's a mistake! Oh no, I made a mistake earlier. \( y = \log x \) (base \( b>1 \)) is an increasing function. So as \( x \) increases, \( y \) increases; as \( x \) decreases, \( y \) decreases. So as \( x \) approaches 0 (decreases towards 0), \( y=\log x \) decreases towards \( -\infty \). So the function is increasing on \( (0, \infty) \)? Wait, no: for \( y = \log_b x \), if \( b > 1 \), it's increasing. So \( y=\log x \) (base 10 or \( e \)) is increasing. So as \( x \) increases, \( y \) increases; as \( x \) decreases, \( y \) decreases. So when \( x \) approaches 0 (decreases), \( y \) decreases (towards \( -\infty \)). When \( x \) approaches \( \infty \), \( y \) increases (towards \( \infty \)).

Now, re - analyzing the options:

1. First option: "The graph has a domain of \( \{x|0 < x < \infty\} \) and approaches 0 as \( x \) decreases."
As \( x \) decreases (gets smaller), \( y = \log x \) decreases. When \( x = 1 \), \( y = 0 \). So as \( x \) decreases towards 1 (from values greater than 1), \( y \) decreases towards 0. But as \( x \) decreases below 1 (towards 0), \( y \) decreases below 0 (towards \( -\infty \)). So the statement "approaches 0 as \( x \) decreases" is true when \( x \) decreases towards 1, but the domain is \( x > 0 \). However, the wording is "as \( x \) decreases" (in general, as \( x \) gets smaller). So maybe the first option is partially correct, but let's check the second option:

2. Second option: "The graph has a range of \( \{y|-\infty < y < \infty\} \) and decreases as \( x \) approaches 0."
The range is correct (all real numbers). And as \( x \) approaches 0 (decreases towards 0), \( y=\log x \) decreases (since it's an increasing function? Wait, no—wait, if it's an increasing function, as \( x \) decreases, \( y \) decreases. So yes, as \( x \) approaches 0 (decreases), \( y \) decreases (towards \( -\infty \)). So this statement is correct.

3. Third option: "The graph has a domain of \( \{x|-\infty < x < \infty\} \) and approaches 0 as \( x \) decreases."
Domain is wrong (logarithm is not defined for \( x \leq 0 \)), so incorrect.

4. Fourth option: "The graph has a range of \( \{y|0 < y < \infty\} \) and decreases as \( x \) approaches 0."
Range is wrong (range is all real numbers), so incorrect.

Wait, but earlier I thought \( \log x \) is increasing, so as \( x \) decreases, \( y \) decreases. So the second option: "decreases as \( x \) approaches 0" is correct (because as \( x \) approaches 0 (decreases), \( y \) decreases). And the range is all real numbers (correct). The first option: domain is correct, but "approaches 0 as \( x \) decreases"—when \( x \) decreases towards 1, \( y \) approaches 0, but when \( x \) decreases below 1 towards 0, \( y \) decreases below 0. So the statement "approaches 0 as \( x \) decreases" is not true in general. So the second option is correct. Also, wait, maybe the first option is correct? Let's take \( x \) values: when \( x = 2 \), \( \log 2\approx0.3010 \); \( x = 1.5 \), \( \log 1.5\approx0.1761 \); \( x = 1.2 \), \( \log 1.2\approx0.0792 \); \( x = 1.1 \), \( \log 1.1\approx0.0414 \); \( x = 1 \), \( \log 1 = 0 \). So as \( x \) decreases from values greater than 1 towards 1, \( \log x \) approaches 0. But when \( x \) decreases from 1 to 0.5, \( \log x \) goes from 0 to \( \log 0.5\approx - 0.3010 \). So the statement "approaches 0 as \( x \) decreases" is true when \( x \) decreases towards 1, but not when \( x \) decreases below 1. However, the domain is \( x > 0 \), so maybe the first option is considering \( x \) decreasing in the interval \( (1, \infty) \) towards 1? The problem's first option says "as \( x \) decreases" (in general, for \( x > 0 \)). So it's a bit ambiguous, but the second option: range is correct, and as \( x \) approaches 0 (which is part of the domain's boundary), the function decreases (since as \( x \) gets closer to 0, \( y \) gets more negative, so it's decreasing). So the second option is correct. Also, let's check the first option again: "approaches 0 as \( x \) decreases"—if we consider \( x \) decreasing (getting smaller) in the domain \( (0, \infty) \), when \( x \) is in \( (1, \infty) \) and decreases towards 1, \( \log x \) approaches 0. When \( x \) is in \( (0, 1) \) and decreases towards 0, \( \log x \) approaches \( -\infty \). So the statement "approaches 0 as \( x \) decreases" is not always true, but the second option's "decreases as \( x \) approaches 0" is always true (since as \( x \) approaches 0 (from the right), \( \log x \) decreases towards \( -\infty \)). So the second option is correct.

Answer:

The correct option is: "The graph has a range of \( \{y|-\infty < y < \infty\} \) and decreases as \( x \) approaches 0." (the second option in the list)