  ## Are You Ready for Calculus?

A solid footing in the skills of precalculus is essential to learning calculus! Specifically, the following skills will be of primary importance:

1. being adept at algebraic manipulation,
2. being familiar with the fundamental results of Euclidean geometry,
3. having a mastery of the trigonometric, inverse-trigonometric, exponential, and logarithmic functions and their properties and relationships, and
4. being able to solve equations involving all of the above

All of the elements above will not be reviewed in any detail here, but I do want to draw your attention to a couple of things that will have particular significance for us as we go forward:

First, I can't emphasize enough how important the notion of a function will be. Recall,

 A function, $f$, is a rule, correspondence, or relationship that maps each element of one set (called the domain) to a unique element from another set (called the range).

There are some things to remember about functions:

• The domain and range don't have to be sets of real numbers. (Although in a first year calculus course, they almost always are.)
• A function can be represented in many ways: as a table of input/output pairs, as a graph, as a formula -- some may even be described verbally.
• For a given function, $f$, we call the range value associated with a given $x$-value in the domain the value of the function $f$ at $x$. This is denoted by $f(x)$, and read as simply "$f$ of $x$"
• When all possible ordered pairs $(x,f(x))$ for a function are graphed on a Cartesian plane with the $x$-axis representing the domain and the $y$-axis representing the range, the graph will pass the "vertical line test". That is to say that any vertical line drawn through an $x$ value corresponding to a domain element will intersect the graph of the function exactly once. This is due to the word "unique" in the definition given above.

Also, when given a formula for a function, one can easily evaluate that function at a given input value through simple substitution. For example, suppose $f(x)=2x^2 - 3$. Then,

$$\begin{array}{rcl} f(a) &=& 2a^2 - 3\\\\ f(-1) &=& 2(-1)^2 - 3 = -1\\\\ f(x+h) &=& 2(x+h)^2 - 3 = 2x^2+4xh+2h^2-3 \end{array}$$

and, using the same principle in a more complicated expression...

$$\begin{array}{rcl} \displaystyle{\frac{f(x+h)-f(x)}{h}}&=&\displaystyle{\frac{(2x^2+4xh+2h^2-3)-(2x^2-3)}{h}}\\\\ &=& \displaystyle{\frac{4xh + 2h^2}{h}} \\\\ &=& 4x + 2h \end{array}$$

Note the last step above, where the $h$ is cancelled, is only valid if $h \neq 0$.

There are many simple functions (square roots, trigonometric and log functions, exponentials, etc...) that one studies in algebra and precalculus. One can build more complicated functions from simpler ones by adding, subtracting, multiplying, dividing, or composing them together.

Recall, composing two functions together is denoted with the symbol "$\circ$" and means to use the output of one as the input of another. For example, if $f(x) = \sin x$ and $g(x) = x^2$, then

$$(f \circ g)(x) = f(g(x)) = f(x^2) = \sin(x^2), \textrm{ and }$$ $$(g \circ f)(x) = g(f(x)) = g(\sin x) = (\sin x)^2 = \sin^2 x$$

Note that above, we wrote $(\sin x)^2$ as $\sin^2 x$. This is a standard convention when dealing with trigonometric functions, but one that frequently causes confusion for students. If the shorter form bothers you, you can always rewrite it using the longer form.

There is a similar trap found in the notation used with logarithms. We write $\ln x$ when we mean $\ln(x)$, dropping the parentheses just like we sometimes do with trigonometric functions. So $\ln x^2$ means $\ln(x^2)$. If one wishes to square the natural log of $x$, one needs to write either $(\ln x)^2$ or $\ln^2 x$ instead.

There are other ways to combine simpler functions to form more complicated ones. We can use exponentiation, for example. Using the same $f(x)$ and $g(x)$ as used in the composition example above, we could define a function $h(x)$ in the following way,

$$h(x)=f(x)^{g(x)} = (\sin x)^{x^2}$$

Additionally, it may be the case that we want to build a function that uses one rule for producing its outputs when one condition applies, and another rule when some other condition applies. Of course we might have several such condition-rule pairs. We call such functions piecewise defined. As an example,

$$f(x) = \left\{ \begin{array}{ccl} 2x^2 &,& x<1\\ 3 &,& x=1\\ x+1 &,& x>1 \end{array} \right.$$

Some functions we say are "one-to-one" or "invertible", if they yield an inverse function, which will map the outputs of the original function back to their respective inputs. Of course this is only possible if both there is a unique output or $y$-value for any given input or $x$-value, and there is a unique input or $x$-value for any given output or $y$-value.

Graphically, such functions pass the "horizontal line test" (in addition to the aforementioned "vertical line test"). That is to say, any horizontal line drawn intersects the graph in at most one point.

(Note, there is no restriction that all of the elements in the range be output elements of the function -- so it is possible a horizontal line drawn could miss the graph of the function completely.)

The inverse function for a given $f(x)$ is denoted by $f^{-1}(x)$, and by its definition we have, for each $x$ in the domain of $f^{-1}(x)$,

$$f(f^{-1}(x)) = x$$

and for each $x$ in the domain of $f(x)$,

$$f^{-1}(f(x)) = x$$

As an example, consider the function $f(x)=e^x$ and it's inverse $f^{-1}(x)=\ln x$. Note that $\ln e^x = x$ for all real values $x$, and $e^{\ln x} = x$ for all $x > 0$

You should always be on watch for opportunities to use these types of inverse relationships to solve equations and simplify expressions. For example, using the properties of logs, we can rearrange the elements of the expression below so that the "$e$" and the "$\ln$" cancel each other out -- simplifying the expression greatly...

$$e^{-\ln x} = e^{\ln x^{-1}} = x^{-1} = \frac{1}{x}$$

Before we leave the subject of inverses, also recall when graphing a function and its inverse, the graphs that result must be symmetric to one another across the line $y=x$.

As an example, consider again the functions $e^x$ and $\ln x$ and their graphs. You do remember what their graphs look like, right?

It will be VERY important as you learn calculus that you can quickly recall how to graph all of the functions seen in precalculus. This includes constant, linear, and quadratic functions, functions involving square roots, circles and semi-circles, piece-wise defined functions, simple power functions, all of the trigonometric and inverse trigonometric functions, and finally, the exponential and logarithmic functions.

Similarly, you should be well versed in graphing transformations of the above functions (Recall: shifting graphs horizontally and vertically, stretching or compressing them horizontally and vertically, and reflecting them across the $x$ or $y$ axes).

As a last point of review, make sure you are familiar with all of the basic trigonometric identities and properties of logarithms.

If a lot of this doesn't sound familiar to you, or you are really confused by the examples or topics above, you might want to consider taking a course more focused on precalculus skills before diving into this course. But -- if you have all of the above firmly under your belt, you should be ready...