Monday 24 February class

(after Test 1)

Topics:

• One-to-one  functions (also called injective functions)

• The horizontal line test for a one-to-one function

• Definition of the inverse function

• Finding a formula for the inverse of a one-to-one function algebraically

Important Notes:

• For a one-to-one function , the inverse function is denoted by . This is read “f inverse of x”. It is not the -1 power! We happen to use the same notation, but when applied to the name of a function it means “inverse” and is not an exponent or power.

• The inverse function reverses the roles of the input and output. Therefore, the domain of  will be the range of  and vice-versa.

• To find the formula of the inverse function, we change x to y and y to x in the formula for the original function, and then solve for y. If we can solve for y (the new y) uniquely in terms of x, then the original function was one-to-one and all is well. If we cannot solve uniquely for y in terms of x, it means that the original function was not one-to-one and so does not have an inverse.

For example, take (which we already know is not one-to-one). If we go ahead and try to find a formula for its inverse, we get this:

(the original function’s formula)

(reversing the roles of x and y)

(I put the term with y on the left, to make it look nicer)

(using the square root principle to solve for y)

So we cannot solve uniquely for y in terms of x after reversing the input and output. This is a reflection of the fact that the original function was not one-to-one.

• As in the example we worked in class, the inverse function will do the opposite (inverse) operations in the reverse order compared to what the original function does.

Homework:

• Reread and review the definitions and the examples worked in class.

• Do the assigned parts of Exercises 7.1 and 7.2

• Make sure that you do the WeBWorK – there is an old one which is extended because there were some problems with it, and also a new, short one. They are due Tuesday (tomorrow!) by 11 PM, so start early!

• Make sure that you do the previously assigned Warm-Up (Warm-Up for Transformations) in Piazza, if you have not already done it. (You can only do it one time.) This is also extended to tomorrow 11 PM because of difficulties that came up in it, but there will be no more extensions!

* Please make sure that you have entered an email address in WeBWorK. I am still getting emails from WeBWorK that I cannot respond to because the students do not have email addresses in WeBWorK. Also, WeBWorK will be used to send out midterm grades later on.

* From now on you should expect that there will be a WeBWorK assignment and a Warm-Up every time (except when there is a Test in the next class meeting), and they will be due by 11 PM the evening before the next class meeting. You should always start early in case there are any difficulties. If you want to discuss how to solve the problems (or homework in general), please use Piazza!

* There is a Piazza app available so you can easily use the Piazza discussion board on your smart phone or iPod touch. See this page for more information.