Question #b8258

2 Answers
Apr 10, 2017

Answer:

The question format is not clear.
Quick answer: log of a negative is not permitted so it is a no no to #x+4<0#

Explanation:

NOT MEANT TO BE: #(fx)=log_2(x+4)-3#
This would be very different!
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#log2# would normally be understood as #log_10(2)# which is a constant.

So we have: #" "f(x)=log_10(2)(x+4)-3#

Set: #f(x)=y=log(2)x+4log(2)-3#

This is the equation of a strait line

So for this condition we have:

domain (input) #->(-oo,+oo)#
range (output) #->(-oo,+oo)#

However; as there is no 'excluded values' in this it is a little disconcerting that mention is made of an asymptote.

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So perhaps the question is meant to be: #(fx)=log_2(x+4)-3#

Set #f(x)=y=log_2(x+4)-3#

Converting this to log base 10 we have

#y=log_10(x+4)/log_10(2)-3#

Now this is different as you do have excluded values, in that, from #log_10(x+4)#

#ul("the values NOT permitted are such that")" "x+4 < 0#

So we have permitted value of

#color(red)(x>=-4" " ->" Domain"->(-4,oo)#

As #log_10(2)<1# then #log_10(x+4)/log_10(2) > log_10(x+4)#

As #x# tends to infinity then the -3 has very little effect so may be discounted

so #color(green)(lim_(x->oo)" "log_10(x+4)/log_10(2)-3=oo" "->"Part of the Range"#

When #x+4# becomes decimal then #log(x+4)# becomes negative. The magnitude of the negative number increases the closer to zero #(x-4)# becomes. Thus :

#color(green)(lim_(x+4->0)" "log_10(x+4)/log_10(2)-3= -oo" "->"Part of the range")#

#color(red)("Range "(-oo,+oo)#

Tony B

Apr 10, 2017

Answer:

See explanation.

Explanation:

The range is the set of all numbers for which the formula is defined.

In this example we have a #log_2# function which (as all logarythms) is defined only for positive values, so to calculate the domain we have to solve:

#x+4>0 #

#x> -4#

So the domain is: #D=(-4;+oo)#

The #log_2# function takes all values, so the range is #RR#

The asymptote is #x=0#, because the closer #x# gets to zero, the smaller is #f(x)#.

#lim_{x->0} f(x)=-oo#

The graph is:

graph{ log(x+4) - 3 [-6, 30, -17.26, 5.25]}