Question

select all the correct answers.
which statements are true about the graph of function f?
f(x) = \log x

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

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Explanation:

Step1: Recall properties of logarithmic function

The function \( f(x)=\log x \) (assuming base 10 or natural log, the domain and range properties are similar). The domain of a logarithmic function \( \log_b x \) (where \( b>0, b eq1 \)) is \( x > 0 \), so in set notation, the domain is \( \{x|0 < x<\infty\} \). The range of a logarithmic function is all real numbers, i.e., \( \{y|-\infty < y<\infty\} \). Also, as \( x \) approaches \( 0^+ \) (decreases towards 0), \( \log x \) approaches \( -\infty \) (so the function decreases as \( x \) approaches 0), and as \( x \) increases, \( \log x \) increases. When \( x = 1 \), \( \log 1=0 \), so as \( x \) decreases towards 0, the function values go to \( -\infty \), but if we consider the behavior as \( x \) decreases (towards 0) from values greater than 1, the function value approaches 0? Wait, no. Wait, when \( x \) is between 0 and 1, \( \log x \) is negative, and as \( x \) approaches 1 from the left (decreases from values greater than 1 to 1), \( \log x \) approaches 0. Wait, maybe the first statement: "The graph has a domain of \( \{x|0 < x<\infty\} \) and approaches 0 as \( x \) decreases." Let's check each option:

1. First option: Domain \( \{x|0 < x<\infty\} \) (correct, since log x is defined for x>0). As x decreases (towards 0, but also, when x decreases from, say, 2 to 1, log x decreases from log 2 to 0. So as x decreases (towards 0, but in the sense of moving left on the x - axis from larger x to smaller x), when x approaches 1 from the right, log x approaches 0? Wait, maybe the wording is a bit ambiguous, but let's check other options.

2. Second option: Range \( \{y|0 < y<\infty\} \) is wrong, because the range of log x is all real numbers (negative, zero, positive). So this is incorrect.

3. Third option: Domain \( \{x|-\infty < x<\infty\} \) is wrong, since log x is not defined for x ≤ 0. So this is incorrect.

4. Fourth option: Range \( \{y|-\infty < y<\infty\} \) (correct, since log x can take any real value). And "decreases as x approaches 0" – as x approaches 0 from the right (x→0⁺), log x→ -∞, so the function is decreasing on its domain (since the derivative of log x is \( \frac{1}{x\ln b} \), which is positive for x>0, wait no! Wait, if \( f(x)=\log x \) (natural log), the derivative is \( \frac{1}{x} \), which is positive for x>0, so the function is increasing on (0, ∞). Wait, I made a mistake earlier. The logarithmic function \( f(x)=\log x \) (base > 1) is an increasing function. So as x increases, f(x) increases, and as x decreases (towards 0), f(x) decreases (since it's increasing, so when x gets smaller, f(x) gets smaller). Wait, let's take base 10: log 10 = 1, log 1 = 0, log 0.1 = -1, log 0.01 = -2. So as x decreases from 10 to 1 to 0.1 to 0.01, f(x) decreases from 1 to 0 to -1 to -2. So the function is increasing? Wait no, when x increases from 0.1 to 1 to 10, f(x) increases from -1 to 0 to 1. So the function is increasing on (0, ∞). So as x approaches 0 (decreases), f(x) approaches -∞ (decreases without bound), and as x increases, f(x) increases. Now, the first option: "approaches 0 as x decreases" – when x decreases towards 1 (from values greater than 1), f(x) approaches 0. So maybe the wording "as x decreases" is in the sense of x moving towards 1 from the right, or in general, when x decreases (gets smaller), if we consider x moving from, say, 2 to 1, f(x) moves from log 2 to 0. So the domain is correct (x>0), and as x decreases (towards 0, but also in the region x>0, when x decreases towards 1, it approaches 0). The fourth option: range is all real numbers (correct), and "decreases as x approaches 0" – since the function is increasing, as x approaches 0 (decreases), f(x) decreases (because x is getting smaller, and the function is increasing, so f(x) gets smaller). Wait, let's re - evaluate:

- Domain of \( f(x)=\log x \): \( x>0 \), so \( \{x|0 < x<\infty\} \) (correct for first and wrong for third).

- Range of \( f(x)=\log x \): all real numbers, \( \{y|-\infty < y<\infty\} \) (correct for fourth and wrong for second).

- Behavior as x decreases (x→0⁺): \( \log x→-\infty \), so the function decreases (since it's increasing, so as x gets smaller, f(x) gets smaller) as x approaches 0.

- Behavior as x decreases (towards 1 from x>1): \( \log x→0 \) (since log 1 = 0).

So let's check each option:

1. "The graph has a domain of \( \{x|0 < x<\infty\} \) and approaches 0 as x decreases." – Domain is correct. As x decreases (for example, from 2 to 1), \( \log x \) approaches 0. So this statement is true.

2. "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 this is false.

3. "The graph has a domain of \( \{x|-\infty < x<\infty\} \) and approaches 0 as x decreases." – Domain is wrong (x must be >0), so this is false.

4. "The graph has a range of \( \{y|-\infty < y<\infty\} \) and decreases as x approaches 0." – Range is correct. As x approaches 0 (decreases), \( \log x \) decreases (since the function is increasing, so smaller x gives smaller f(x)), so this statement is true.

Step2: Confirm the correct options

After analyzing the domain, range, and behavior of the logarithmic function \( f(x)=\log x \), we see that the first option (domain \( \{x|0 < x<\infty\} \) and approaches 0 as x decreases) and the fourth option (range \( \{y|-\infty < y<\infty\} \) and decreases as x approaches 0) are correct. Wait, but earlier I thought the function is increasing, but as x approaches 0, f(x) decreases (because x is getting smaller, and the function is increasing, so f(x) gets smaller). For example, x = 0.5, f(x)=log 0.5 (negative), x = 0.1, f(x)=log 0.1 (more negative), so as x approaches 0 (decreases), f(x) decreases (becomes more negative). And the first option: when x decreases from, say, 2 to 1, f(x) decreases from log 2 to 0. So "approaches 0 as x decreases" – when x decreases towards 1, it approaches 0. So both first and fourth options are correct? Wait, let's check the original options again:

- First option: "The graph has a domain of \( \{x|0 < x<\infty\} \) and approaches 0 as x decreases." – Domain is correct. As x decreases (in the domain x>0), if we consider x moving towards 1 from the right (x>1, decreasing towards 1), f(x) approaches 0. So this is true.

- Fourth option: "The graph has a range of \( \{y|-\infty < y<\infty\} \) and decreases as x approaches 0." – Range is correct. As x approaches 0 (decreases), f(x) decreases (since the function is increasing, so smaller x gives smaller f(x)), so this is true.

The second option has wrong range (range is not y>0), third option has wrong domain (domain is not all real numbers).

Answer:

The correct answers are:

• The graph has a domain of \( \{x|0 < x<\infty\} \) and approaches 0 as x decreases.
• The graph has a range of \( \{y|-\infty < y<\infty\} \) and decreases as x approaches 0.