Determine the function which corresponds to the given graph. a natural logarithmic function crossing the y axis at zero and going through the point 2,1.The asymptote is x = -1.

Accepted Solution

The general natural logarithmic function [tex]y=ln(x)[/tex] has an asymptote at [tex]x=0[/tex]. To make this function have an asymptote at [tex]x=-1[/tex], we need to translate the function 1 unit left. So the new function (according to rules of translations) takes the form[tex]y=ln(x+1)[/tex]This function should also go through two points [tex](0,0)[/tex] and [tex](2,1)[/tex]. Let's check (0,0).[tex]ln(0+1)=ln(1)=0[/tex]. So the first point is okay. Let's check (2,1).[tex]ln(2+1)=ln3[/tex], which is not equal to 1.A simple trick to make this equal to 1, for us to satisfy all the conditions of making this function, is to think "What can we do to [tex]ln(3)[/tex] to make it equal to 1?"Simple! We have to multiply it by [tex]\frac{1}{ln(3)}[/tex]!Now, our final equation becomes [tex]y=\frac{1}{ln(3)}ln(x+1)[/tex].ANSWER: [tex]y=f(x)=\frac{1}{ln(3)}ln(x+1)[/tex]