 # Domains and Inverse Functions

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Domains and Inverse Functions
Sections 1.7 and 1.8

Objectives Determine the domain and range (where possible) of a function given as an equation. Determine if a function given as an equation is one-to-one. Determine if a function given as a graph is one-to-one. Algebraically find the inverse of a one-to-one function given as an equation.

Objectives State the domain and range of a function and it inverse.
State the relationships between the domain and range of a function and its inverse Restrict the domain of a function that is not one-to-one so that an inverse function can be found. Draw the graph of the inverse function given the graph of the function.

Vocabulary inverse function horizontal line test function composition
one-to-one function

Domain Questions Does the function have a denominator?
Does the function have a square or even root? Does the function have a log or ln in it? Did the function arise from finding an inverse? Is this a “real world” problem?

Find the domain of the function:

Find the domain of the function:

Find the domain of the function:

Find the domain of the function:

Given the functions and find each of the following:

Determine if the function is one-to-one.

Steps for finding an inverse function.
Change the function notation f(x) to y. Change all the x’s to y’s and y’s to x’s. Solve for y. Replace y with f -1(x).

Find the inverse of the function
Find the domains of the function and its inverse.

Find the inverse of the function
Find the domains of the function and its inverse.

Find the inverse of the function
Find the domains of the function and its inverse.

Find the inverse of the function
Find the domains of the function and its inverse.

Draw the graph of the inverse function for the graph of f(x) shown below.

The function is not one-to-one
The function is not one-to-one.  Choose the largest possible domain containing the number 100 so that the function restricted to the domain is one-to-one.  Find the inverse function for that restricted function.