We are asked to determine
the domain, range, and equation of the vertical asymptote.
We are given G of X equals
the opposite of log base three of X plus four.
Let's begin by reviewing the definition of a log function,
as well as the properties of a log function.
Where Y equals log base B of X is a log function,
where both X and B are greater than zero,
and B can't be one.
The equivalent exponential equation
for Y equals log base B of X
is B raised to the power of Y equals X.
The domain is the open interval from zero to infinity
because X must be greater than zero.
Notice how X is the input into the log function.
The range is all real numbers or the open interval
from negative infinity to positive infinity.
The vertical asymptote is X equals zero,
and the horizontal intercept is the point one comma zero.
So for example to look at the graph
of the basic log function Y equals log base three of X,
notice how this function does have all of these properties.
And now looking at G of X, we have G of X equals
negative log base three of X plus three,
so because we have log base three of X here
the domain, or set of all possible X values,
is the same as a basic log function,
which is X greater than zero
or using a interval notation,
the open interval from zero to infinity.
And the vertical asymptote is also X equals zero,
and then the range for the basic log function
is all real numbers or the open interval
from negative infinity to positive infinity.
So looking at G of X, if we change
the sign of the function value and then add four,
this won't change the range,
the range is still all real numbers
or the open interval from negative infinity
to positive infinity.
And now let's verify these results graphically.
One way to graph G of X is to recognize
that G of X is a transformation of the basic log function
Y equals log base three of X.
Where if we let F of X equal the basic log function
with base three, we could write G of X in terms of F of X,
as saying G of X is equal to the opposite of F of X,
and then plus four.
Where this negative sign here would reflect
the graph of F of X across the X-axis, the horizontal axis,
and then the plus four would shift the graph up four units.
Let's verify this with an animation.
In red we have the graph of Y equals log base three of X,
and now we will graph Y equals A
times log base three of X plus D,
or notice when A is one,
and D is zero we have the graph of the same log function
which is now graphed in blue.
And now if we change A to negative one
this will give us the graph of Y equals
the opposite of log base three of X,
which is a reflection of Y equals log base three of X
across the X-axis, which we see here.
And now if we change D to positive four
we will have the graph of G of X,
and notice how this will shift the graph up four units.
So this blue graph is a graph of the given function G of X.
So going back to our work,
I already have the graph here on the right,
you'll notice how there is a vertical asymptote
at X equals zero, which is the y-axis.
Notice as X approaches zero from the right
the function values approach infinity
verifying the vertical asymptote.
We can also see the domain is when X is greater than zero,
or the open interval from zero to infinity,
and then for the range, notice how the graph goes up
as well as down indefinitely.
And therefore this does verify
the range is the open interval
from negative infinity to positive infinity.
It may appear as if there is a horizontal asymptote here
at Y equals 2, but there's not,
this graph does go down forever
even though it goes down very slowly.
I hope you found this helpful.