Inverse Functions, Logarithms, and Derivatives of Logarithmic and Exponential Functions

Inverse Functions, Logarithms, and Derivatives of Logarithmic and Exponential Functions

MAT1241: Calculus 1
University of Eswatini


Inverse Functions
An inverse function “undoes” the effect of another function. If we have a function (f(x)), its
inverse is denoted as . Here are the key points:

1. Definition of Inverse Function:
○ A function (f) has an inverse if and only if it is one-to-one (i.e., each output
corresponds to a unique input).
○ The inverse function swaps the roles of input and ou
for all valid inputs.
2. Finding the Inverse Function:
○ To find the inverse of (f(x)):
1. Replace (f(x)) with (y).
2. Swap (x) and (y): .
3. Solve for (y): .
3. Properties of Inverse Functions:
○ The graph of is the reflection of the graph of (f) across the line (y = x).


Logarithms
1. Definition of Logarithm:
○ The logarithm of a positive number (x) with base (b) is denoted as .
○ It represents the exponent to which (b) must be raised to obtain (x): .
2. Common Logarithm and Natural Logarithm:
○ Common logarithm:
○ Natural logarithm:
3. Logarithmic Properties:
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