Question 1130365
 if you have this:

 {{{y = log(3,(x-2)) + 4 }}}

domain - The domain of {{{y}}} is the set of all {{{x}}} values such that {{{x - 2 > 0 }}}
Solve the above inequality to obtain the domain: {{{x >  2 }}}
{ {{{x}}} element {{{R}}}:  {{{x >  2 }}} }

The range of {{{y}}} is given by the interval:

 ({{{- infinity }}}, {{{ infinity}}}).


b.
 The vertical asymptote is obtained by solving the equation: {{{x - 2 = 0 }}}
which gives {{{x =  2 }}}
As {{{x }}}approaches {{{2}}} from the right ({{{x > 2}}}) , {{{y}}} decreases without bound because there is a vertical asymptote. How do we know this?



c.
 - To find the {{{x}}} - intercept we need to solve the equation {{{y = 0}}} or 
{{{log(3,(x -2))+4 = 0}}} 
{{{log(3, (x -2)) = -4}}} 

Rewrite the above equation in exponential form:
 
{{{x - 2 = 3^-4}}}

{{{x - 2 = 1/3^4}}}

{{{x - 2 = 1/81}}}

{{{x  = 1/81+2}}}

{{{x  = 1/81+162/81}}}

{{{x  = 163/81}}}

The {{{x}}} intercept is at the point  
The {{{y}}} intercept is at the point ({{{0}}} , {{{y(0)}}}) 

=> ({{{0}}} , {{{log(3, (0 -2))+4}}}) =>{{{log(3, (0 -2))+4}}} have complex solution, there is no {{{y}}} intercept 

d - So far we have the domain, range, {{{x}}} and {{{y }}}intercepts and the vertical asymptote. We need more points to draw a graph. 


table:

{{{163/81}}} | {{{0}}}

{{{2.5}}} | {{{3.4}}}

{{{4}}} | {{{4.6}}}

{{{5}}} | {{{5}}}

{{{6}}} | {{{5.3}}}

{{{8}}} | {{{5.6}}}

{{{10}}} | {{{5.89}}}

plot points and draw a graph:


{{{drawing ( 600, 600, -10, 10, -10, 10,

circle(163/81,0,.14),circle(4,4.6,.14),circle(5,5,.14),circle(6,5.3,.14),
circle(8,5.6,.14),circle(10,5.89,.14),line(2,-10,2,10),circle(2.5,3.4,.14),
graph( 600, 600, -10, 10, -10, 10, log(3,(x-2)) + 4 )) }}}