|Real Number System |||Absolute Value |||Functions and Graphs |||Trigonometry |||Basic Graphing Skills |||Studying a Function |||Algebraic Skills |||Answer Section|
|Section 1 | Section 2 | Section 3 |||Section 4 | Section 5 |||Section 6 | Section 7 | Section 8 | Section 9 | Section 10 | Section 11 | Section 12 |||Section 13 | Section 14 | Section 15 | Section 16 | Section 17 | Section 18 |||Section 19 |||Section 20 | Section 21 | Section 22 |||Section 23 |||Answers|
During this year you will need to use the idea of inverse functions, so let's review some of the basic concepts you should remember.
How is an inverse function formed?
An inverse function is formed by exchanging x for y.
If y = 3x + 2, then the inverse function would be x = 3y + 2 would be the inverse function. But you can solve this equation if you would like for .
When does a function have an inverse function?
A function must pass the horizontal line test to have an inverse function. This is necessary because the definition of a function requires that for every x there exists exactly one y. Remember the same y value can be assigned to more than one x value in a function.
If an original function has two x values which yield the same y value, then when the x and y values are switched there will be two y values for the same x value. This would prevent the switched equation from being a function. Therefore, it is necessary to restrict the domain on some functions so that they will pass the horizontal line test.
Suppose . Recall that this function does not pass the horizontal line test. x=1 and x = -1 both yield the same y value of 1. It is necessary to restrict the domain of the function so that it passes the horizontal line test. You have two choices. Either define it as or . I have selected the first choice. Now when the x and y are switched the equation becomes Notice this new graph is a reflection of the original graph over the line y = x.
Check your understanding: (Remember to write down your answers.)
1. Find the inverse for the functions:
2. Find the inverse for each of the following functions after restricting the domain so the function passes the horizontal line test.