Did you read my tips on how to study and learn Math 1500?  If not, here is a link to those important suggestions:

These are tips for the first assignment in the Distance/Online Math 1500 course, but I strongly recommend that you do this assignment as homework even if you are taking the classroom lecture section of the course.  These assignments are very good (and challenging) practice.  The first assignment is a great way to build and review key skills that will be helpful for this course.

Here is a link to the actual assignment, in case you don't have it:

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

Don't have my book? You can download a sample containing those two lessons here:

The assignment also delves pretty deep into some high school algebra that I have not discussed in my book, so I will endeavour to give you some more pointers when relevant.

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.  Make sure you sub the number in place of every x in the function to establish the sign of the function in that region.

Do not try to find range algebraically!  As my example in question 3 of Lesson 1 demonstrates, you can still miss key facts about the range even if you do your algebra correctly.  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.
• 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 to get a feel what the limit is.  In that case, add about three numbers to the table of values.  For example, if you want to see what the function is doing as x approaches infinity, add three increasing values to your table, such as 10,000 and 100,000 and 1,000,000.  Remember, you are allowed to use a calculator on your assignments, so feel free to do so while computing the y coordinates for these values.  If you notice the y value is getting closer and closer to some number as x approaches infinity, you have discovered there is a horizontal asymptote on the far right of the graph as x approaches infinity.  Do the same if negative infinity is in your domain.  Add -10,000, -100,000, and -1,000,000 to your table and see if y approaches a horizontal asymptote on the far left of the graph.
• If your domain has an endpoint that causes a bottom zero, it is possible that you have a vertical asymptote as x approaches that number on your graph.  Compute the limits as x approaches that value from the left side and the right side to see what the graph is up to.  Or, use your calculator again to investigate y's behaviour as x gets closer and closer to the given value.  For example, if you discovered that x=2 is causing a bottom zero in your function, add three points to your table of values that are getting closer to 2 on each side.  I start with a value that is 0.1 away from 2.  So, if I want to investigate what the graph looks like as I approach 2 from the left (numbers smaller than 2), I would start 0.1 to the left at 1.9, then come closer with 1.99 and 1.999.  Sub those three values into the function and see what is happening to y as you get closer to 2 (perhaps it is getting larger and larger towards infinity or more and more negative towards negative infinity).  Then, approach 2 from the right, starting at 2.1 (0.1 larger than 2), then come closer to 2 with 2.01 and 2.001.
  
• Plot all those points and connect the dots to get a reasonably reliable picture of the function.  You can now visualize the range, by scanning the graph from the very bottom (where y is negative infinity) to the very top (where y is positive infinity) to see what y values are included in the range.
• Be sure to include your graph in your answer to the question.  A graph is always sufficient evidence to justify your answer for domain and range (provided the graph is correct, of course).
  

I show you how to find domain and range in Lesson 1 of my book and my "easier" tips above.  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.

Don't worry about how long you may be spending making a table of values.   The questions on this assignment are intentionally challenging because they know you have the time and access to a calculator.  If you have to spend ten minutes plotting ten or so points and drawing a graph, go ahead.  That's what mathematicians do.  Don't ever think you are supposed to just know what the graphs look like.  Nor, will you try to memorize such things.  Rest-assured, you will learn some very useful skills as this course progresses to sketch complex graphs more quickly, but there is no shame in making a table of values to learn about the behaviour of a function.

Note that an inequality is just another way of asking if something is positive (above 0) or negative (below 0).  Just make a sign diagram here to find the solution.

I briefly discuss Even and Odd functions in Lesson 9 (Curve-Sketching) while answering question 3 part a.

• A function is even if         f(-x) = f(x).
• A function is odd if         f(-x) = - f(x).

Which means you need to substitute "-x" in place of each x in the function and simplify.  If your answer is identical to the orginal function, f(x), that proves the function is even.  If your answer is almost identical to the orginal function, but you would have to multiply your answer by -1 to make it identical to the orginal function, you have shown that it is the same as -f(x), and therefore an odd function.  If it is not identical to f(x) or -f(x), then it is neither.

Tip: Compute f(1) and f(-1) for each function.  If you get the same answer for both, that probably means the function is even.  If you get the same answer, but opposite signs, that probably means the function is odd.  Check again, this time using f(2) and f(-2).  Once you have an idea whether a function is even, odd or neither, then proceed to prove it by computing f(-x).  Note that subbing in 1 and -1, 2 and -2, proves nothing.  That is just work to do on your own if you want to get an idea.  To prove your answer properly, you must compute f(-x) and compare the result to f(x).

For part (c), draw a sketch of the y = sinx curve and you can see from that whether the function is odd, even or neither.  Google "sine curve" if you don't know what the graph looks like.

I show you how to find the inverse of a function in Lesson 8 of my book (just after question 4 in the lecture, page 230).  I like to immediately have x and y change places, then proceed to isolate y.  Hint: to isolate y, collect the terms that have y on the left side of the equation, and move the other terms to the right.  Then factor y out.  Once you have isolated y, make sure you replace y with f^-1 (x), the f-inverse symbol.

I think there is a typo in this question.  Contact the prof.  I believe they mean cube root x and fourth root x for the two terms, but, as written, they are saying ln is raised to the power of 3 and it is just a square root of x, then ln is raised to the power of 4 for a square root of x.  They might mean that, but it is much more boring.  If they actually meant "cube root of x" and "fourth root of x", note that you can use fractional exponents as taught in Lesson 5 of my book (page 118).  Then this simplifies very nicely using the Log Laws I teach at the end of Lesson 1.

This is actually preparing you for the preliminary steps to solve Max/Min problems taught in Lesson 10 of my book.  Label the length and width of your rectangle x and y, respectively.  You can then come up with an equation for the area, and sub that into the equation for the perimeter.  Look at my question 1, in Lesson 10 and see what I do for my constraint equation and Q equation, as an idea of what you are supposed to do here.  Don't even try to understand any more than that at this stage.

These are called 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".

This is a good run-through of limits.  Study Lesson 2 thoroughly to prepare for this question.  Hint: if I let u = square root of x, what would x-squared be?  That might help you see how to factor part (d).

Read the Squeeze Theorem section at the end of the lecture in Lesson 2 of my book.  They have given you a pretty easy one.

To sketch this piecewise function, make a table of values for each piece (be sure to choose x values that are relevant to each piece).  Be sure to include the endpoints for each piece.  If a particular piece does not actually include the endpoint (for example, the first piece is for x < 0, so it is up to but not including 0), evaluate the endpoint anyway, but plot a hole instead of a dot (so sub x=0 into the first piece to see the coordinates of the hole for that first piece).

Once you have drawn the graph, solve the limits by reading your graph as I instruct with my graph-reading exercise at the start of Lesson 2.