Note that you need to study Lesson 1 (Skills Review) and Lesson 2 (Limits) from my Intro Calculus book to prepare for this assignment.

Question 1 requires the Vertical Line Test.  If a vertical line can cross more than one point anywhere on the curve, the graph is not a function. 

 

It is sufficient to say, "It is a function because it passes the Vertical Line Test," or "It is not a function because it fails the Vertical Line Test."

 

Question 2.  I show you how to find domain and range in Lesson 1 of my book.  Remember, any x value that causes the denominator to be zero (a Bottom Zero) must be excluded from the domain.  Also, if there is a square root, use a sign diagram to determine when the sign under the square root is negative since that, too, must be excluded from the domain.

 

Sign Diagrams: Many students have trouble understanding my steps for making a sign diagram in Lesson 1.  Here is an easier method to remember (although it is slower for most students, it is perfectly adequate for this stage of the course).  Once you have found the Top and Bottom Zeros and marked them on the number line, pick a number from each region and substitute it into the entire function.  The sign of your answer will be the sign of that region.  For example, if you have marked the numbers 1 and 4 on the number line, choose a number less than 1 (such as 0) to sub into the function to find the sign of that region, choose a number between 1 and 4 (such as 2) to sub into the function to find the sign of that region, and choose a number greater than 4 (such as 5) to sub into the function to find the sign of that region.

 

Range: A safer way to find the range of a function is to draw a graph of the function and then visualize the range that way.  Find the domain of the function first.  Put the endpoints of the domain on a Table of Values and add about three more points in between the endpoints as well.  Plot those points and connect the dots to get a reasonably reliable picture of the function. 

 

Make sure you have read the asymptote section in Lesson 2 (leading into question 15).  If the domain includes infinity or negative infinity, compute the limit as x approaches infinity and negative infinity to see what is happening to the function (and so y) at that time.  You can also try some numbers near infinity and negative infinity, like 10,000 or 1 million to get a feel what the limit is.  Remember, you are allowed to use a calculator on your assignments, so feel free to do so while making the sign diagram. 

 

Similarly, if you have excluded x values from the domain because they are bottom zeros, do limits as x approaches those bottom zero values to learn what the graph is doing as you approach that region.  Again, you can use numbers near these bottom zeros to confirm what you have learned from the limit.  For example, if you have found that 2 is a bottom zero, compute the limit as x approaches 2 from both the left side and the right side to see what the y value (and graph is doing).  Verify this on your table of values, by subbing in numbers like 1.9 and 1.99 as well as 2.1 and 2.01. 

 

Include your sketch of the curve in your solution to back up your conclusion about the domain and range. 

 

You have plenty of time on these assignments, so they intentionally give some challenging questions.  Take the time to draw a good sketch, plotting 10 points if you have to.  Then it is merely a matter of reading the graph from bottom (where y is negative infinity) to top (where y is positive infinity) to see what y values are included in the domain.

 

Question 3 is composite functions.  f o g is f(g(x)), telling you to sub g(x) in place of x in the f function.  Conversely, g o f is g(f(x)).   Note the domain of f o g can be no better than the domain of the "inside" function (g in that case).  Establish the domain of g, then establish the domain of "f o g" as well and whichever domain is smaller is the domain of f o g.  Similarly, the domain of "g o f" is the smaller of the domains of f and the domain of "g o f".

 

Question 4.  I show you how to find the inverse of a function in Lesson 8 of my book (just before question 4 in the lecture).  Note that, since, to invert a function, x and y change places, that means the graph of f ^-1 is basically the graph of f "turned sideways" (the x-axis becomes the y-axis and vice-versa).  This also means the domain of f is identical to the range of f ^-1 and the domain of f ^-1 is the range of f. It is much easier to find domain than range.  Find the domain of f, that tells you the range of f^-1, the inverse.  Then, of course, once you have found the inverse function find its domain.

 

A one-to-one function is one where each value of x has only one value of y (of course, that is true of all functions), but also each value of y has only one x.  Graphically speaking that means all functions pass the Vertical Line Test, but one-to-one functions also pass the Horizontal Line Test.  One way to answer part (a) in this question is to graph the function and show it passes both the Vertical and Horizontal Line Tests.  But the graph isn't easy and make take several points.  It also requires you check for Vertical and Horizontal Asymptotes by doing all k/0 limits and limits as x approaches negative and postive infinity.

 

To prove a function is one-to-one algebraically, you do the following:

 

Step 1:

Define f(x₁) by subbing x₁ in place of x and define f(x₂) by subbing x₂ in place of x.

Step 2:

Write this definition down in your proof you will submit:

"A function is one-to-one if f(x₁) =  f(x₂) if and only if x₁ = x₂."

Step 3:

Set  f(x₁) =  f(x₂) and use algebra (like cross multiplication and multiplying everything out to get rid of brackets, etc.) to collect all the terms with x₁ on one side of the equation and all the terms with x₂ on the other side of the equation.

Step 4:

If everything goes according to plan, all the terms disappear, leaving you with "x₁ = x₂".

Step 5:

Assuming that does happen, you can declare you have proven f(x) is one-to-one.

 

It is sufficient to say, "The inverse is a function because it passes the Horizontal Line Test," or "The inverse is not a function because it fails the Horizontal Line Test."

 

Question 6 is just like question 4.  Part (b) is especially challenging.  Hint: be sure to review the log and exponential section of Lesson 1 in my book.  At some point you will need to use "ln", the natural log to help isolate y.

 

Question 7 is similar to my example right at the start of Lesson 2.  Follow the path of the graph as you approach each limit.  What y value does the path lead you to?

 

Question 8 requires the various tricks and tips I teach in Lesson 2.  

 

Question 9 obviously requires the Squeeze Theorem, which I teach in Lesson 2, question 17.