# Arithmetic and Algebra

1. Expand the product $(a+b)^7$ where $a$ and $b$ are variables representing unknown real numbers.

2.  Here is an expression involving the variable $x$:

$\displaystyle \frac{1}{1+x^2}$

(a) Is the expression $(1+x^2)^{-1}$ equal to the original one?  Why or why not?

(b) Is the expression $1^{-1} + x^{-2}$ equal to the original one?  Why or why not?

(c) Is the expression $\displaystyle \frac{x}{x^3+x}$ equal to the original one?  Why or why not?

(d) Is the expression $\displaystyle \frac{1+x}{1+x^2+x}$ equal to the original one?  Why or why not?

(e) Are there any numbers that we are not allowed to plug in for $x$?  If so, what are they?  If you don’t think there are any restrictions on $x$, explain why.

3.  Is it always true that $\displaystyle \left(x^{1/2}\right)^2 = x$?  Try a few examples (like $x = 1, 2, -1$, etc.).

4. Find values of $a, b,$ and $c$ so that

$\displaystyle x^3 - y^3 = (x-y)(ax^2+bxy + cy^2)$

or explain why no values can be found.

5.  Multiply out (i.e. expand) the expression $(\theta^2-\theta+1)(\theta+2)$.  (Note: why did I use the variable name $\theta$?  No reason!  Just trying to change things up!!)

6.  (a) What do you get when you compute $(-1)^0$?

(b) What do you get when you compute $(-1)^{1}$?

(c ) What do you get when you compute $(-1)^{2}$?

(d) What do you get when you compute $(-1)^{3}$?

(e) What do you get when you compute $(-1)^{-3}$?

(f) What do you get when you compute $(-1)^{2014}$?

(g) In general, if $n$ is a positive, whole number, describe (in words, in formulas, or both)    what happens when you compute $(-1)^n$.

(h) What about $(-2)^n$?

(i) Why didn’t I ask you about $(-1)^{1/2}$?

(j) What do you get when you compute $(-1)^{1/3}$?

7. Find numbers that you can plug in for $a$ and $b$ so that the equation

$\displaystyle (a+b)^4 = a^4 + b^4$

is true.  How would you explain to a student learning algebra that your example does not, in fact, mean the above equation is always true?

8$^*$.  Express the following sum as a single fraction:

$\displaystyle 1 + \frac{1}{4} + \frac{1}{9}.$

9$^*$.  Express the following product as a single fraction:

$\displaystyle \frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}.$

10. I decided to waste money on a self-proclaimed psychic, and he gave me a terrible reading — it was vague and inaccurate.  I told him (rather angrily) that I doubted he was a true psychic, and, to his credit, he decided to prove to me that he was legit.

Here’s what he told me: “Write down any number.  No, seriously.  Any number, and don’t show it to me.  Okay.  Now, I want you to square that number and write down the result.  Great.  Next add one to the squared-number you just wrote down.  Cool.  Finally, take your final expression and raise it to the zero power.”  He then paused for dramatic effect, finally announcing, “I know what your final number is, even though I never knew the number you started with.”

Low and behold, he wrote down the exact number I had produced.  So, what happened?  Was I in the presence of a genuine psychic?  Please help find an alternative explanation so that I am not tempted to waste any more money on such services!  Also, what number did he show me I had produced?

# Functions

1.  A formula-definition for a function usually looks something like this

$f(x) = \text{mathy, algebraic expressions involving } x.$

Write down a formula for a function — any function! — that produces the outputs $f(1) = 1, f(0) = 0$ and $f(2) = 4$.  (Note: there are LOTS of correct answers to this problem, so you should compare what you’ve come up with to what your friends thought about.)

2.  Explain why there is no function that satisfies $f(1) = 1$ and $f(1) = 4$.

3.  Unfortunately, functions are not allowed to eat any real number input.  For example, the function $f(x) = x^{-1}$ is not allowed to have as an input $x = 0$.  Why not?

(a) The function $f(x) = x^{1/2}$ also has “problems” at certain input values for $x$.  What values of $x$ can we not use as an input for this function?

(b) What inputs are and are not allowable for the function $f(x) = \sqrt{1-x}$?

(c ) How about the function $g(x) = \frac{1}{1+x^2}$ (this should look familiar)?  What inputs are and are not allowable for this function?

4.  Sketch a graph of the function $f(x) = 3x+1$, plotting the out-put values along the $y$-axis as we discussed in class.  Of course, there are far too many input values for $x$ to plug in (in fact, there are infinitely many), so you probably want to plot just a few points, like $(-1, f(-1)), (0, f(0)), (1/3, f(1/3)) (1/2, f(1/2)), (1, f(1)),$ and, say, $(2, f(2))$.  Based on the location of these points, make a(n educated) guess about what the whole graph for this function looks like.

5.  Recall from your previous algebra and pre-calc courses that a line (in the plane) can always be thought of as the graph of a function $f(x) = mx + b$ where $m$ and $b$ are fixed numbers.

(a) What is the number $m$ called?  Actually, its okay if you’ve forgotten the official or technical name for this number.  What’s more important is that you think about this question: how does the value $m$ affect our line?

(b) What sorts of lines are obtained when $m = 0$?

(c ) What is the number $b$ called?  Similarly, its okay if you’ve forgotten the name of this quantity, too.  What’s more important is this: What does this number have to do with our line?

(d) If I tell you that my line passes through the points $(1,2)$ and $(4,5)$, what must the numbers $m$ and $b$ be?

(e) There’s one type of line that cannot be written down as the graph of a function $f(x) = mx + b$.  What kind of line is it?