How do you graph, find the intercepts and state the domain and range of #f(x)=6^x+3#?

1 Answer
May 30, 2017

Answer:

y-intercept #->(x,y)=(0,4)#

No x-intercept

domain #color(white)(.)->{x: x in (-oo,+oo)}#

range #" "->{y: y in (+3,+oo)}#

Explanation:

Set #" "y=6^x+3#

#color(blue)("Determine the y-intercept")#

Set #x=0# giving

#y=6^0+3 = 4#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine the x-intercept")#

Set #y=0=6^x+3#

#6^x=-3#

Take logs of both sides

#xln(6)=ln(-3)#

But ln(-3) is undefined thus there is no x-intercept
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine end behaviour "x->-oo)#

#lim_(x->-oo) 6^x+3 = lim_(z->+oo) 1/6^z+3 ->k = 3#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine end behaviour "x->+oo)#

#lim_(x->+oo) 6^x+3" " ->" "k+3 =oo+3=oo#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Tony B