MA1375 Precalculus Spring2014 TTh

The WeBWorK and Warm-Up deadlines from last class have been extended to Monday evening 11 PM.

Topics:

• 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.

• The domain of is the range of , and the range of is the domain of .

The “round-trip theorem”:

If f and g are one-to-one functions, then they are inverse to each other if and only if both of the following are true:

for all x in the domain of g

AND

for all x in the domain of f.

This theorem provides a way of checking if two functions are inverse to each other.

• The relationship between the graphs: the graph of  is the reflection of the graph of  over the line y=x. (This is because they reverse the roles of y and x.)

• For a function which is not one-to-one: Restricting the domain to an interval where the function is one-to-one, so that we can find an inverse on the restricted domain. Note that when we restrict the domain, we want to make sure that the entire range of the function is covered in the restricted domain.

One example we discussed is . Here is a bit more detail than I gave in class.

If we restrict the domain to the interval , then f(x) is one-to-one on that interval and every value in the range of f is reached for some value of x in the restricted domain. [This would not be true if, for example, we restricted to the open interval . Then f would be one-to-one on the restricted domain, but the value 0 in the range of f would not be in the new range of the restricted f, because no x-value in the new domain gives f(x)=0.]

On that restricted domain, the inverse function is . We can see this by the algebraic method:

(remember that we are taking  when we restrict the domain)

Reverse x and y:  (and now )

Solve for y: 

– We only take the positive branch of the square root, because  due to the restriction on the domain of f(x). So there is a uniquely defined inverse function for f(x) on this restricted domain.

What if we restricted the domain to the interval  ? On this interval f is one-to-one (this is just the left-hand side of the graph) and the entire range of f is covered. But in this case, the inverse function on the restricted domain would be . Why is this so? Because when you try to find the inverse algebraically,

(remember that we are taking  when we restrict the domain)

Reverse x and y:  (and now )

Solve for y: 

– We now only take the negative branch of the square root, because  due to the restriction on the domain of f(x). So there is a uniquely defined inverse function for f(x) on this restricted domain, but it is different from the inverse we get if we restrict the domain as we did the first time.

Moral of the story: there may be more than one choice for how to restrict the domain of a function which is not one-to-one, but different choices will lead to different inverse functions.

• Polynomials: definitions and important vocabulary you should know. (See the list below)

• Long division of polynomials.

Next time we will learn a shortcut efficient method for doing long division of polynomials in certain cases. But there are still times when we will need to use the full long division, so make sure that you practice it!

Vocabulary you need to know for polynomials:

• monomial

• polynomial

• term

• coefficient

• degree (of a term, or of a polynomial)

• constant term – note that the degree of a constant term is defined to be 0.

• leading term and leading coefficient

• root of a polynomial (= zero of the polynomial function = x-intercept of the graph of the polynomial)

• Related to division: dividend, divisor, quotient, remainder

Homework:

• Reread and review the definitions and examples discussed in class. Make sure that you know and can use all the vocabulary correctly!

• Do the assigned parts of Exercises 7.2-7.4, and 8.1

• Take a look at the “Review of functions and graphs” which follows Session 7. (But skip #1.7, which we did not discuss, unless you want to try it – it’s not hard.)

• Do the WeBWorK: due by Wednesday 11 PM extended to Monday 11 PM. Start early! and use the Piazza discussion board if you get stuck on any of them (or on the regular homework)!

• Read ahead in Session 8 and then do the Warm-Up for Remainders – due also by 11 PM Wednesday  extended to Monday. (This one uses a Google Docs form instead of Piazza. Please only submit your answer ONE TIME. You should print or save the receipt that you get when you have submitted it.)

Please note that from now on you are expected to do all the online assignments on time and to inform me promptly if there is any difficulty doing them. I have been very flexible so far, but it’s time to get serious.  I believe we have had time to shake out all the bugs in the system. There will be no more re-assignments or extensions except under very unusual circumstances. So get to it!

Topics:

• Reminders about the relationship between the square root and squaring.

• Composition of functions: Definition 6.5 p. 77. Start here!

• Finding the formula of a composite function

• Using functions defined by tables

• The square root principle (see this post) vs. the definition of square root: what is ? (Came up in Example 6.8(c)) – see fuller discussion below

• One-to-one functions

• The horizontal line test for one-to-one functions

The square root principle and related things:

The square root principle (as it is sometimes called) says that

if

then .

This is related to two other facts connecting squaring and square roots:

• — This is the definition of the square root of x. ( is the number which, when squared, gives you x.)

• — This is because the radical sign indicates that you should take the non-negative square root, so  is always greater than or equal to 0 no matter what x is.

Many people get the bad idea in their heads that  is the same as x. That is not true as a general rule. It is only true when . Be careful!

Homework:

• Reread/ review the examples. Make sure that you understand how composition works: we go through first one, then the other, of the functions.

• Do the assigned parts of Exercises 6.3, 6.4, 6.5, and 7.1

• Do the WeBWorK: start early! post questions to Piazza!

• Do the Warm-Up   (due by Monday night 11 PM also)

See the flyer below…

MathSpecialistHelpNYCCTSpring2014

(after Test 1)

Topics: (some review from last time)

• Basic graphs: the graphs of the following functions are essential to know:

Important feature: this graph has an inflection point (the s-curve) at (0,0)

Important feature: the domain is only the interval

(I forgot to mention this one today!)

Important feature: the graph approaches the lines x=0 and y=0 (the two coordinate axes). These lines are called asymptotes. Notice that x=0 is not in the domain.

• Transformations of graphs:

vertical translation (shift) – add a number to the output (value of the function)

horizontal translation (shift) – add a number to the input

reflection in the x-axis – multiply the output by -1

reflection in the y-axis – multiply the input by -1

[There are two transformations we did not discuss:

vertical stretching or squeezing – multiply the output by a positive number

horizontal stretching or squeezing – multiply the input by a positive number

We will return to these when we need them.]

• Even and odd functions:

Basically, even functions have graphs which have the same symmetry as the graph of and odd functions have graphs which have the same symmetry as the graph of . The algebraic tests are tests for those symmetries.

Algebraic tests:

A function is even if and only if for all x in its domain.

A function is odd if and only if for all x in its domain.

Homework:

• Reread/ review all the examples, making sure that you see how the transformations change the graphs (and why!) and how the algebraic test for an even or odd function checks the symmetry. You can use Piazza to discuss if you have questions.

• Do from Session 5: Exercise 5.1 (all), 5.2 (a-e), 5.5 (a-e)

• 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.

• Make sure that you do the WeBWorK – there are two old ones which are extended because there were some problems with their dates. They are due Thursday ( day after 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.

Test 1 is scheduled for Tuesday 25 February, the first 50 minutes or so of class. Please review the course policies: in particular, there are no make-up tests and no extra time for latecomers, so please be early!

I handed out review Self-Tests in class, and I will also post them to Piazza. You can use Piazza to discuss these problems or to tell me if you find any typos!

The answers follow, after the fold. Please use the Piazza Discussion Board to discuss these!

Continue reading →

Topics:

• Functions defined by graphs: reading the domain and range from the graph

• Basic use of graphing calculator

• Using the “zeroes” and “maximum”, “minimum” calculate functions

• Some important graphs of basic functions

• Vertical and horizontal shifts (translations) of basic graph

Homework:

• Go through the examples using the graphing calculator, following the instructions to see how it works.

• Reread/review the Examples 3.8-3.12

• Finish the assigned problems in Session 3

• Do the assigned problems in Session 4

• Do the WeBWorK – due by 11 PM Monday the 17th! Extended to Wednesday the 26th 11 PM. Start early!

• Do the Warm-Up on Piazza. You will have to join Piazza if you have not already done so. Make sure that you then join the TTh group in Piazza, so you can see the questions and polls for this section.

Topics:

• Expressing one variable as a function of another

Examples 2.20, 2.21, 2.22

• The square root principle (used in Example 2.22 in class):

When solving an equation of the form (something), we use the Square Root Principle:

If , then

$latex x = \pm\sqrt{c}

Note: there is no “missing step”. We do not “take the square root of both sides”. What is wrong with doing that has to do with the fact that is NOT necessarily x. (And yes, I know that some teachers tell you to “square root both sides”, ugh, but it is still wrong, and will cause you trouble later on.)

Here is a video that explains it correctly. (You’ll have to sit through an annoying ad first, but the video is good and is totally mathematically correct.)

If you write the incorrect version, it is an error (that’s the main consideration!) and will cause you to have fewer points on that problem (if that’s all you care about). So correct this in your thinking if necessary!

• Functions given by formulas

Example 3.1: computing various values of the function, and thinking about the domain of the function (what values cause difficulties? Where is the function defined?)

We went ahead and found the domains for each of the functions in Ex. 3.1

(a) A polynomial function 3x+4:The domain is R

(b) A radical function : we need the radicand to be nonnegative  ,  so the domain is found by solving an inequality. We solved it the same way we solved absolutevalue inequalities before: the domain is

(c) A piecewise-defined function: we find the domain by fitting together the various intervals where the function is defined

(d) A rational function: we need the denominator not to be 0, so the domain is all real numbers except -3

Example 3.5: finding the domains of more functions of these four types.

• Computing values of the function for more complicated inputs, and computing a difference quotient.

Example 3.2(e-k)

See also Example 3.3(a)

Homework: see the previous post.

I’m working on today’s post. Since I expanded the discussion of most of the examples today, I’m trying to type up notes of that expansion for you. So I hope you will be patient while I do that!

In the meantime…

•Homework from the course outline:
The assigned parts of Exercises 2.7, 2.8

In Session 3: the assigned exercises up to Exercise 3.6

• Do the WeBWorK assignment: it is due Wednesday evening 11 PM, so start early!

• Make sure to join our course in Piazza. If you responded to the first Warm-Up (Exercise 1.1) you should have received an invitation to join. Otherwise, follow the link on the “Resources” page. Also make sure to join the TTh group in that course. It is vital that you join: the Warm-Ups will mostly be in Piazza from now on! Also, you can use Piazza to discuss any questions you have about homework problems. I have already posted a question there for you to contribute to the answer, to get practice using Piazza. Homework points are given to students who contribute to the answers!

• The Warm-Up is here. Due by 11 PM Wednesday! (Please also note that this is not on Piazza, but from now on we will be using Piazza for the Warm-Ups, as indicated above.)