Question

select all the true statements about the function ( y = -ln x ).
the function has a ( y )-intercept at ( (0,1) ).
the function has an ( x )-intercept at ( (1,0) ).
the function has a vertical asymptote at ( x = 0 ).
the function has a horizontal asymptote at ( y = 0 ).

Explanation:

Step1: Check y - intercept

For y - intercept, set \(x = 0\). But \(\ln(0)\) is undefined. So the statement “The function has a y - intercept at \((0,1)\)” is false.

Step2: Check x - intercept

For x - intercept, set \(y=0\). Then \(0 =-\ln x\), which implies \(\ln x=0\). Since \(\ln x = 0\) when \(x = e^{0}=1\). So the function has an x - intercept at \((1,0)\).

Step3: Check vertical asymptote

The domain of \(y =-\ln x\) is \(x>0\). As \(x
ightarrow0^{+}\), \(\ln x
ightarrow-\infty\), so \(y =-\ln x
ightarrow+\infty\). Thus, the function has a vertical asymptote at \(x = 0\).

Step4: Check horizontal asymptote

As \(x
ightarrow+\infty\), \(\ln x
ightarrow+\infty\), so \(y=-\ln x
ightarrow-\infty\). There is no horizontal asymptote at \(y = 0\).

Answer:

The function has an x - intercept at \((1,0)\) and the function has a vertical asymptote at \(x = 0\).