the graph of y = ln x and the corresponding points that lie on the graph of f(x)=2 + ln x. points that lie…

Explanation:

Step1: Find x - value for y = - 1 in y = ln x

Set ln x=-1. By the definition of the natural - logarithm, if ln x = a, then x = e^a. So x = e^(-1)=\frac{1}{e}.

Step2: Find the corresponding point on f(x)=2 + ln x

Substitute x=\frac{1}{e} into f(x). f(\frac{1}{e})=2+\ln(\frac{1}{e})=2 - 1 = 1. The point is (\frac{1}{e},1).

Step3: Find x - value for y = 0 in y = ln x

Set ln x = 0. Since ln x = 0 when x = e^0=1.

Step4: Find the corresponding point on f(x)=2 + ln x

Substitute x = 1 into f(x). f(1)=2+\ln(1)=2+0 = 2. The point is (1,2).

Step5: Find x - value for y = 1 in y = ln x

Set ln x = 1. Then x = e^1=e.

Step6: Find the corresponding point on f(x)=2 + ln x

Substitute x = e into f(x). f(e)=2+\ln(e)=2 + 1=3. The point is (e,3).

Answer:

Points that lie on the graph of y = ln x	Corresponding points that lie on the graph of f(x)=2 + ln x
(\frac{1}{e},-1)	(\frac{1}{e},1)
(1,0)	(1,2)
(e,1)	(e,3)