Notes 1.5 – Inverse and Logarithmic Functions
There are several ways to graphically, numerically, and algebraically determine the inverse of a relationship.
Algebraically –
Graphically –
Numerically –
Find the inverse of each of the examples below. Include the domain and range for the original function and the
inverse. Also determine if the inverse is also a function.
f ( x)  ( x  1)2  3
g ( x)  x  1  4
h( x)  ( x  2)3
There are several ways to know if the inverse of a function will also be a function.
Graphical Test: How do you tell if a relationship is a function?
So what would make sense to call the graphical test to know if the inverse is a function?
One-to-One: every x goes to a unique y and every y goes to a unique x
Which of the previous functions is one-to-one?
If the inverse is a function, we can use inverse function notation.
For example, since the inverses of g(x) and h(x) are functions, then the inverses can be defined using the
notation g 1 ( x) and h1 ( x) .
In order to make the inverse of some relationships be a function sometimes you must restrict the domain of the
original function.
Look at the function f(x) from the previous page. Is the inverse of f(x) a function? Is f(x) one-to-one?
So in order for the inverse to be a function we must restrict the domain of f(x).
Example:
What would you restrict the domain of f(x) so that the inverse is a function?
Graph the restricted function of f(x) and f-1(x).
What is the domain and range of each?
In order for two functions to be inverses of each other, the input of one function must pair with the output of
the other function and the output of one function will match to the input of the other function.
Example:
In order to prove that two functions are inverses of each other, show that the composite of the functions is equal
to the identity function.
Example: Are the following two functions inverses of each other?
Now complete the assigned inverse function worksheet.