Problems 64-67 are based on material learned earlier in the...

Problems 64-67 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
For $f(x)=7^{x-1}+5$, find $f^{-1}(x)$.

Key Concepts

Inverse Functions

Inverse functions reverse the roles of the input and output of the original function, meaning that if f(a)=b then f?¹(b)=a. Finding an inverse typically involves swapping the dependent and independent variables and solving for the new dependent variable, which requires a firm understanding of algebraic manipulations.

Exponential Functions

Exponential functions are characterized by a constant base raised to a variable exponent. These functions often model growth or decay and have unique properties such as a non-zero domain and a range that is always positive. Manipulating these functions typically requires techniques like applying logarithms in order to invert them.

Logarithmic Functions

Logarithms are the inverse operations of exponentiation. They are essential for solving equations where the variable is an exponent. Understanding logarithms allows one to convert an exponential equation into a form that can be solved by isolating the variable using properties of logarithms.

Function Transformations

Function transformations involve modifications of the basic function through operations such as translations, scaling, or reflections. Recognizing transformations is important when finding an inverse, as shifts and stretches of the graph can affect how the original input and output are related, yet the process of swapping variables for the inverse remains fundamentally similar.

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