Here are some tips to help you with Assignment 1 if you are taking Math 1500 via distance/online.

 

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

 

Question 2.  I show you how to find domain and range in Lesson 1 of my book.

 

Many students have trouble understanding my steps for making a sign diagram in that lesson.  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.

 

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 inbetween the endpoints as well.  Plot those points and connect the dots to get a reasonably reliable picture of the function.  If the domain includes infinity or negative infinity, compute the limit as x approaches infinity or negative infinity, respectively, to see what is happening to the function (and so y) at that time.

 

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 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).  You also need to study my Logs and Exponentials section of Lesson 1 to help find the inverse in this question.  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.  Note, when finding the domain of a logarithmic function, you can only log positive values (you can't log 0, you can't log a negative).

 

Question 5.  Since f ^-1 is the f graph turned sideways, for a function to be invertible as a function also, it must pass the Horizontal Line Test as well as the Vertical Line Test.

 

Question 6 uses the same concepts I discussed above in question 4.

 

Question 7 is similar to my example right at the start of Lesson 2.

 

Question 8 requires the various tricks and tips I teach in Lesson 2.  Note that x^1/3 is the cube root of x.  Hint: Let u = cube root of x, then what would x itself be in terms of u?

 

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