91Ó°ÊÓ

An inverse function essentially reverses the effect of the function it is associated with. Given a function \( f(x) = \ln(\ln x) \), its inverse function \( f^{-1}(x) \) will take an output value from \( f \) and return the original input \( x \). To determine this inverse, we start by rewriting the function equation \( y = \ln(\ln x) \).

To isolate \( x \), you first undo the outer logarithm by exponentiating both sides, resulting in \( e^y = \ln x \). Then, undo the inner logarithm in a similar way: \( x = e^{(e^y)} \).

This outcome \( f^{-1}(x) = e^{(e^x)} \) confirms how the inverse reverses \( f \). With this formula, if you apply \( f \) to a value, applying \( f^{-1} \) retrieves the original input, completing the cycle.

The natural logarithm, often notated as \( \ln(x) \), is a logarithmic function to the base \( e \). The constant \( e \) (approximately 2.718) is a mathematical constant used widely in growth calculations, particularly in natural processes. The function \( \ln(x) \) gives the power that \( e \) needs to be raised to, to get \( x \). For example, \( \ln(e) = 1 \), because \( e^1 = e \).

When calculating \( \ln(\ln x) \), you are essentially applying the natural log twice. The inner \( \ln x \) must be positive, meaning \( x > 1 \). Then, for the outer \( \ln \) to be defined, \( \ln x \) itself must be greater than zero, resulting in \( x > e \). This makes the domain of \( f(x) = \ln(\ln x) \) be \( x > e \). The natural log's properties like base \( e \) and positivity are key in defining the allowable inputs.

Function composition involves combining two or more functions such that the output of one function becomes the input of another. In the context of \( f(x) = \ln(\ln x) \), the composition starts with \( g(x) = \ln x \), then applies \( h(y) = \ln(y) \) to \( g(x) \). This composition is represented as \( f(x) = h(g(x)) \).

The process can be thought of in layers: \( g(x) = \ln x \) first processes the input \( x \), transforming it into \( \ln x \). Then, the function \( h(y) = \ln(y) \) further processes the output of \( g(x) \), resulting in \( \ln(\ln x) \).

Function composition is notated as \( (h \circ g)(x) \), indicating \( h \) is applied after \( g \). Understanding function composition helps in analyzing how complex functions are built from simpler ones, as seen in this exercise.