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Graphs are useful because many people prefer looking at pictures to looking at equations. They're not necessarily better, though. While a picture may be worth a thousand words, it's really only worth one equation.

When giving a presentation, a graph makes a very nice visual. Different sorts of graphs show up in real life, from president approval rating surveys, to financial market summaries, so it's good to be comfortable reading them. Slip into your sock monkey slippers, stoke the fire and settle in for an evening of satisfying graph-reading.

Graphs let us see patterns in numbers that we might not be able to see merely by looking at a list. Lists are nice some of the time, but are we running out to the store for a gallon of milk right now? We didn't think so.

We can look at the relation generated by the equation *x* = *y*, or we can look at a graph of a straight line. Lots of things in real life, such as population sizes or the height of something thrown into the air, can be measured to generate graphs with familiar shapes, including straight lines, quadratic functions, and exponential functions. Remember last week when you never would've counted quadratic functions or exponential functions among a list of "familiar shapes"? Move over, triangles and squares. There are some new shapes in town.

### I Like Abstract Stuff; Why Should I Care?

Set theory is a branch of mathematics that lets mathematicians make math even more abstract than it already is. It's like the Jackson Pollock of mathematics. In set theory, not even the numbers 0, 1, 2, 3, 4, . . . are taken for granted. They're built from scratch, just like your grandma's raisin oatmeal cookies.

Starting with only the curly brackets and a comma, we can build the whole numbers. Here's how it works. Pretend 0, 1, 2, 3, 4, . . . are symbols, but we don't know what they mean yet. We have to define them. First, we define 0 be the empty set:

0 = { }

Then, we define 1 to be the set containing 0:

1 = {0}

Then, 2 as the set containing 0 and 1:

2 = {0, 1}

See where this is going? Each whole number can be defined as the set of all whole numbers that came before it:

*n*= {0, 1, . . . , (*n*– 1)}From here, there are ways to define the rest of the numbers (integers, rational numbers, and real numbers, for example) and the arithmetic operations, all in terms of sets. Unfortunately, set theory can't help you to define your calves. Sorry, you're going to need to go to the gym. There's no way around it.

There are many different kinds of sets, and some sets have other sets contained within them as elements. However, there's no such thing as a set containing all the sets there are, since that one biggest set wouldn't be able to contain itself. Did we blow your mind yet? This concept is known as Russell's Paradox, and is often presented as a story about a barber. This site gives a nice presentation of Russell's Paradox and explains the connection between the story and the set theory.

### How to Solve a Math Problem

There are three steps to solving a math problem.

- Figure out what the problem is asking.

- Solve the problem.

- Check the answer.

These steps let us solve problems we haven't seen before, which is good, or else we'd be in trouble on exams.

### Sample Problem

Graph the relation described by the inequality

*y*≤ 4 – 2^{-x}*.*This is similar to graphing a linear inequality, except the inequality isn't linear. Here, we need to ask: "Whose Non-Linear Inequality Is It, Anyway?"

So what do we do? Let's go through our problem-solving steps.

**1. Figure out what the problem is asking.**Well, the problem says to graph something. We'll probably need to graph

*y*= 4 – 2^{-x}along the way, so we need to figure out what that looks like first. Then, since we're being asked to graph an inequality, we'll need to figure out what part of the graph to shade in. We might use our charcoals for this, just to be extra-artistic.

**2. Solve the problem.**First, let's graph

*y*= 4 – 2^{-x}.This is an exponential function. We could re-write the equation as

*y*= (-1)2^{-x}+ 4which looks a little more like the exponential functions we worked with earlier. Familiarity sometimes breeds contempt, but in this case it breeds happiness, since this will make it easier to solve. Let's do the easy bit first. The

*y*-intercept is:*y*= (-1)2^{0}+ 4 = 3Since the constant term is 4, the asymptote is

*y*= 4.Now comes the fun part, as if you weren't already rolling on the floor with uncontrollable belly laughs already. Since the exponent is -

*x*instead of just*x*, the exponential curve will be turned upside-down. Because the exponential term is being multiplied by a negative number, the curve will also be turned left to right. Putting everything together, the graph looks like this:We now have a graph of

*y*= 4 – 2^{-x}, which is very nice, but not what the problem asked for. We're supposed to be graphing an inequality, which means we'll need to do some shading. We know you're a rebel at heart, but no coloring outside the line on this one. To determine where we need to shade, let's think about it like a linear inequality. The points we want are the points on this curve, or those points where*y*is*less*than it would be on the curve. This means we want to shade in the lower portion of the graph:**3. Check the answer.**To check our answer, let's take one point in the shaded area and make sure it should be included in the relation, then take one point

*not*in the shaded area and make sure it shouldn't be included in the relation. Then let's be sure to keep the two separated, since they'll invariably start fighting any time they get within five feet of each other.First, the point (3, 0) is currently in the shaded portion of the graph. Should this point really be in the relation? When

*x*= 3, the right-hand side of the inequality is.

The value

*y*= 0 is certainly less than or equal to , so yes, this point should indeed be included.The point (-4, 0) is not in the shaded portion of the graph. When

*x*= -4, the right-hand side of the inequality is4 – 2

^{{-(-4)}}= 4 – 16 = -12.The value

*y*= 0 is certainly not less than -12, so the point (-4, 0) is not in the relation. While we can't check infinitely many points, checking that these two came out on the correct sides of the inequality is reassuring. If you're a perfectionist, however, and are dead set on checking infinitely many points, good luck. We'll check back in the fall and see how you're doing.- Figure out what the problem is asking.