# Functions

### Topics

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} + 4

which 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

(-1)2^{0} + 4 = 3.

Since 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 will 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 is

4 – 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.