Question Video: Finding the Domain of the Inverse of an Exponential Function Mathematics

Consider the function ๐‘“(๐‘ฅ) = ๐‘^๐‘ฅ, where ๐‘ is a positive real number not equal to 1. What is the domain of ๐‘“โปยน(๐‘ฅ)?

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Video Transcript

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

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 ๐‘ฅ is equal to ๐‘ to the power of ๐‘ฅ, the inverse function is equal to log base ๐‘ of ๐‘ฅ. 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 ๐‘ฅ is greater than zero as the only values we can substitute into the function log base ๐‘ of ๐‘ฅ are ๐‘ฅ greater than zero.

An alternative method here would be to consider the graphs of our functions. The graph of ๐‘“ of ๐‘ฅ is shown. It intersects the ๐‘ฆ-axis at ๐‘ and the ๐‘ฅ-axis is an asymptote. The inverse of any function is its reflection in the line ๐‘ฆ equals ๐‘ฅ. This means that the function log base ๐‘ of ๐‘ฅ intersects the ๐‘ฅ-axis at ๐‘ and the ๐‘ฆ-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 ๐‘ฅ 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 ๐‘ฅ 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 ๐‘ฅ is all values greater than zero. This once again proves that the domain of the inverse function is ๐‘ฅ is greater than zero.

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