### Video Transcript

Consider the function ๐ of ๐ฅ is
equal to ๐ to the power of ๐ฅ, where ๐ is a positive real number not equal to
one. What is the domain of the inverse
of ๐ of ๐ฅ?

There are a few ways of approaching
this problem. One way would be to recall that
exponential functions and logarithmic functions are the inverse of each other. This means that if ๐ of ๐ฅ is
equal to ๐ to the power of ๐ฅ, the inverse function is equal to log base ๐ of
๐ฅ. We are asked to find the domain of
this function. The domain of any function is the
set of input values. We know that we can only find the
logarithm of positive values. This means that the domain of the
inverse function is ๐ฅ is greater than zero as the only values we can substitute
into the function log base ๐ of ๐ฅ are ๐ฅ greater than zero.

An alternative method here would be
to consider the graphs of our functions. The graph of ๐ of ๐ฅ is shown. It intersects the ๐ฆ-axis at ๐ and
the ๐ฅ-axis is an asymptote. The inverse of any function is its
reflection in the line ๐ฆ equals ๐ฅ. This means that the function log
base ๐ of ๐ฅ intersects the ๐ฅ-axis at ๐ and the ๐ฆ-axis is an asymptote. As the domain is the set of input
values, we can see from the graph that the domain of the inverse of ๐ of ๐ฅ is all
numbers greater than zero.

A final method would be to recall
that the domain of ๐ is equal to the range of the inverse. Likewise, the range of ๐ of ๐ฅ is
equal to the domain of the inverse. The range of any function is the
set of output values. We can see from the graph that the
range of ๐ of ๐ฅ is all values greater than zero. This once again proves that the
domain of the inverse function is ๐ฅ is greater than zero.