Determine the domain, range,
and equation of the vertical asymptote
of g of x equals negative natural log
of the quantity x minus six.
Let's begin by reviewing the definition and the properties
of a basic log function.
Where a log function is in the form of y equals
log base b of x where both x and b are greater than zero
and b can't be one.
The log equation y equals log base b of x
is equivalent to the exponential equation
b raised to the power of y equals x.
The domain is the open interval from zero to infinity,
again because x has to be greater than zero.
The range is all real numbers
or the open interval from negative infinity
to positive infinity, because in exponential notation,
the value of y can be any real number.
The vertical asymptote is x equals zero
and the horizontal intercept is the point one comma zero.
And if we look at the graph
of the basic natural log function shown here,
notice how it does have all of these properties.
But notice for g of x, we have natural log
of x minus six, not natural log x,
and therefore to find the domain,
we need to solve the inequality x minus six
greater than zero.
Adding six to both sides gives us x is greater than six,
which is the domain of g of x,
which using interval notation is the open interval
from six to infinity.
Instead of setting x equal to zero,
we set x minus six equal to zero.
So if we set x minus six equal to zero and solve,
we get x equals six, which is the equation
of the vertical asymptote.
Which should make sense because notice how the domain
is the open interval from six to infinity.
And then the range of a basic log function
is all real numbers,
and notice how for g of x, we have this negative sign here,
but by changing the sign of the function values,
we don'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 review these results graphically.
One way to graph g of x is to recognize
that g of x is a transformation of the basic
natural log function.
Where if we let the basic natural log function
be equal to f of x,
we could say that g of x is equal to the opposite
of f of the quantity x minus six,
where the negative sign here is this negative sign
and the input into the function f is x minus six.
In this form, we should be able to recognize
this negative sign would reflect the graph
of the basic natural log function across the x-axis
or horizontal axis and then the input of x minus six
would shift the graph right six units.
Let's verify this with an animation.
In red we have the graph of the basic natural log function
y equals natural log x.
And now we will graph y equals a times natural log
of the quantity x minus c,
where if a is one and c is zero,
we get the same graph of the basic natural log function,
which is now graphed in blue.
And now if we change a to negative one,
we will have the graph of y equals
negative natural log x, which will reflect the graph
of the basic natural log function across the x-axis,
which we see here.
And then if we change c to negative six,
notice how it then shifts the graph right six units,
giving us the graph of g of x,
which is y equals negative natural log
of the quantity x minus six.
So this blue graph is a graph of g of x,
which I've already graphed here on the right.
We'll notice how we do have a vertical asymptote
at x equals six.
Notice graphically, as x approaches six from the right,
the function values approach infinity,
verifying the vertical asymptote.
We could also see the x values of the domain
is x greater than six or the open interval
from six to infinity.
We can also verify the range is all real numbers
or the open interval from negative infinity
to positive infinity
because the graph goes up and down indefinitely
without any holes or breaks.
I hope you found this helpful.