Welcome to Precalculus. This page is a guide to the Precalculus topics that are blogged about on this site.

## Functions, generally

Precalculus is mostly about functions, so it makes sense to explore what a function is. A function is a relation between two things, call them *x* and *y,* for which every *x* has one and only one corresponding *y*. For example, in a relation if *x* is the year and *y* is the high temperature for the year in a certain place, then this relation is a function, because each place should have just one high temperature for a year. On the other hand, if *x* is, again, the year and this time *y* is all the temperatures recorded in a certain place, then that relation is not a function, because there will be many temperatures recorded over the course of a year, so each value of *x* has more than one value of *y*.

On a graph, a function is a relation that passes the vertical line test. That means that any vertical line that crosses the graph will cross at one point only. That’s a quick way to identify a function. Here are some examples of applying the vertical line test.

A function’s domain is the set of values *x* can take and its range is the set of values *y* can take. This post shows an example of how to find a function’s range.

## Polynomial functions

One class of functions that you may see a lot of in Precalculus is polynomials.

When you study polynomials, you are typically interested in where they cross the *x*-axis, when they change from increasing to decreasing and vice versa, and what happens at very large values of *x* or *-x*. That last issue is called *end behavior*.

Think back to when you studied quadratics. You might get an equation like What’s the first step in solving it? Get everything of substance onto the left-hand side. Then the right-hand side equals 0. After that, if you can, factor the left-hand side and set each factor equal to 0. You can also solve some higher-degree polynomial equations by factoring, but it gets more challenging as the equations get more complicated. So we find some rules to help us.

### Where does the graph cross the x-axis?

A *x*-value where a graph crosses the *x*-axis can be called an *x*-intercept or a root or a solution (because it represents the solution to an equation). Often your job will be to find those *x*-values.

How many *x-*intercepts do you need to look for? The maximum number zeros of a polynomial function is equal to the function’s degree. What are those *x*-values? The rational roots test can give you some clues.

### End behavior

When *x* or *-x* gets very large, does a polynomial’s value approach negative infinity or positive infinity? That depends on the function’s degree and on the sign of its leading term.

### Other issues

You may be given solutions and asked to find the function that has those solutions.

## Rational functions

In Precalculus you’ll also study rational functions, which are fractions with variables in their denominators.

A rational expression may have vertical, horizontal, or slant (aka oblique) asymptotes. An asymptote is a line that a graph approaches but never reaches.

- A zero denominator may cause a vertical asymptote.
- If the numerator and denominator have equal degrees, there is a horizontal asymptote.
- If the numerator’s degree is less than the denominator’s, there is a horizontal asymptote at
*y*=0. - If the numerator’s degree is exactly one more than the denominator’s, there is a slant asymptote.
- If the numerator’s degree is more than one more than the denominator’s, there is no asymptote.

A polynomial function of *x* divided by a binomial is an example of a rational function, for example . The remainder theorem says that when a polynomial function of *x* is divided by *x-c*, where *c* is any real number, the quotient equals the function’s value at point *c*. With the remainder theorem you can use division — that is polynomial long division or synthetic division — to find a function’s remainder at a point; you can also use it to find a function’s value at a point from its remainder there. For example, to find the value of the function at *x*=-5, divide by x+5. That will give you a remainder of 24. Thus *f*(-5) = 24.

## Exponential Functions

*Exponential growth *is an increase in something — bacteria, money, population, whatever — that is in proportion to the amount present. Here is one example of exponential growth. Here is another.

But things don’t always increase; sometimes they decrease. If the decrease is in proportion to the amount present, that’s exponential decay. Here is an example of exponential decay

An *exponential equation *is an equation with variables in one or more exponents. Here are some thoughts about how to solve an exponential equation.

## Logarithmic functions

A logarithm is kind of a funny animal; it doesn’t always do what you might imagine. This is illustrated in the graph of the logarithm of a product.

## Sliding a function sideways

Let’s say you’re looking at the graph of a function *f*(*x*). Then what does *f*(*x*+1) look like? Imagine yourself standing at the origin of the coordinate plane, and say the function at *x*=o, where you are, is above your head and you are looking up at it. And you wonder what’s happening one unit in front of you, at *x*=1. You, at *x*=0, are looking at the *y*-value for *x*=1. Since you’re at x=0, you replace the y-value that’s already there (in green) with the *y*-value you found for *x*=1 — that is, with *f*(1) (in blue).

What if you do that for every *x-*value on the graph: At every point, replace the *y-*value with the *y-*value you see for *x*+1. Then, at every point, you are replacing *f*(*x*) with the y-value that is one point to the right. Thus, *f*(*x*+1) is one unit to the left of f(x), no matter how much that plus sign makes you want to move it to the right. Similarly, *f*(*x*-1) is *f*(*x*) moved one unit to the right. See more about horizontal translations here. See more about horizontal translations here.