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

select all the correct answers.\nwhich statements are true about the graph of function ( f )?\n( f(x) = log x )\n- the graph has a domain of ( {x|0 < x < infty} ) and approaches 0 as ( x ) decreases.\n- the graph has a range of ( {y|-infty < y < infty} ) and decreases as ( x ) approaches 0.\n- the graph has a range of ( {y|0 < y < infty} ) and decreases as ( x ) approaches 0.\n- the graph has a domain of ( {x|-infty < x < infty} ) and approaches 0 as ( x ) decreases.

select all the correct answers.\nwhich statements are true about the graph of function ( f )?\n( f(x) = log x )\n- the graph has a domain of ( {x|0 < x < infty} ) and approaches 0 as ( x ) decreases.\n- the graph has a range of ( {y|-infty < y < infty} ) and decreases as ( x ) approaches 0.\n- the graph has a range of ( {y|0 < y < infty} ) and decreases as ( x ) approaches 0.\n- the graph has a domain of ( {x|-infty < x < infty} ) and approaches 0 as ( x ) decreases.

Answer

Explanation:

Step1: Analyze domain of $f(x)=\log x$

Logarithmic functions are only defined for positive real numbers, so domain is ${x|0 < x < \infty}$.

Step2: Analyze range of $f(x)=\log x$

Logarithmic functions output all real numbers, so range is ${y|-\infty < y < \infty}$.

Step3: Analyze behavior as $x\to0^+$

As $x$ approaches 0 from the right, $\log x$ decreases to $-\infty$, so the function decreases here.

Step4: Analyze behavior as $x$ decreases

As $x$ decreases towards 0, $\log x$ approaches $-\infty$, not 0.

Step5: Evaluate each option

  • Option1: Incorrect (does not approach 0 as $x$ decreases)
  • Option2: Correct (matches range and behavior as $x\to0$)
  • Option3: Incorrect (range is not ${y|0 < y < \infty}$)
  • Option4: Incorrect (domain is not all real numbers)

Answer:

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