Functions

Calculus deals with the study of Functions and their behavior. So, it's very important that the student have a good understanding of these important class of curves.

I mentioned that functions are special curves, so what is a curve? A curve is nothing more than a visual for an infinite grouping of points (for most cases) in a given pattern. Remember, each point is an ordered pair of numbers, first from the X-axis, 2nd from the Y-axis.

There is usually a rule (equation) that dictates how the x and y coordinates are related. For example, if the y coordinate is always twice the x coordinate, the set of points will line up in a definite pattern. Here are some of them: (1,2), (2,4), (3,6), ....,(100,200),....,(-20,-40), & so on. There will be an infinite number of these ordered pairs that will determine the pattern of the curve. In this case, all of these points will lie on a straight line determined by the rule (equation), y=2x. This will be the equation of this line. So, in this case, y=2x describes a curve which is a straight line.

We can get a visual of this line by plotting (placing a dot at many points determined by the equation) on the XY plane. This is called a graph of the curve determined by its equation.

A different equation would, most likely, produce a different kind of line or curve.

For example, y=3x-5 would produce different ordered pairs but would still line up straight. So, it will be an equation of a different line. However, y=x^2 (x squared) would not produce a straight line. It would be curved up with its lowest point at the orgin. This curve is called a Parabola. Fortunately, we have calculators that will give us a graph of any equation we enter into the input. A great time-saving device.

For the serious math student who looks into this more deeply, they will notice that the XY coordinate system is not convenient nor possible for locating points on some curves. So, other types of movements are invented that make it much easier or possible for analyzing different types of curves. For example, instead of traveling left or right then up or down to locate a point, it may be more convenient to rotate a certain number of degrees (think of the orgin as a little circle) about the origin then going outward a given distant. This is the basis of a new coordinate system based on "polar coordinates". So, each order pair would not have an x and y, but instead a distance out and an angle rotation. These will be the polar coodinates for the curve. Many very interesting looking curves are developed in the polar system, many of which are very difficult to describe or impossible to get in the XY system. So, when someone mentions a different coordinate system, just think of it as a different path of locating points on the curve. In multivarible calculus (Calculus III) there are several that are developed and used.

Back to our XY system, certain curves are Functions and some are not. So, in short, what makes a curve a Function. Keeping it in simple terms for the beginner, just think of a curve that will satisfy the vertical line test. That is, if all vertical lines that cut the curve do so in exactly one point only, it's a Function. This will eliminate circles, ellipses, and many other curves from the class of functions. The serious math student will be exposed to other methods and ways of determining the class of Functions & techniques that can be used to study non-functional curves.
Another simple way of looking at this property will be with ordered pairs. If verial lines cut our curve in exactly one point only, then there will never be two or more ordered pairs (points on the curve) on the same vertical. Say it another way, the graph of a function will never have two points on its graph with the same first element (x coordinate with different y-coordinates). For example, (7,3) and (7,5) cannot be on the graph of a function (these points are on the same vertical line where all x's are 7). Which, in turn, illustrates a vertical line cutting the graph in more than one point.

Functional Notation: Special notation is used when working with functions. Let's go back to our equation y=2x which determined a straight line when graphed. In this equation, the y coordinate will always be twice the x coordinate. Since this curve determines a function, special notation is is used. We say, y is a function of x. In symbols, y=f(x). This does not mean f times (x). It's just a way of stating that this curve is a function. It does not give the equation unless it is stated what the relationship is specifically. If I say f(x)=2x, then I know the functional equation. The important feature is that f(x) is taking the place of y. We could state all the ordered pairs on this function as (x,f(x)). So, y and f(x) have the same meaning. However, by using the functional notation, it becomes very convenient when the equation is in the form f(x)=2x. That way, we can input any x and get instant results. For example, for this function, I can say f(0)=2(0)=0, which means that when x=0, y=0. I can describe the ordered pairs in just one line equation instead of two. I can say quickly, f(1)=2, f(5)=10, f(-4)=-8, & so on. Much more concise then saying when x=-4, y=-8. Of course, you always have the option of not using functional notation and going back to the, so called, y notation. But always remember, when using functional notation, f(x) is replacing y.
Let's take a few more examples:

f(x)=3x²-5x+4, f(0)=4 (when x=0 is substitued, the y-value or functional value is 4). f(1)=3(1)²-5(1)+4=2, f(-1)=3(-1)²-5(-1)+4=12 (tricky).

f(x)= 4/(x²-4), f(0)=4/-4=-1, f(1)=4/(1²-4)= -4/3. Note: x=2 or x=-2 cannot be used since it creats a zero in the denominator (bottom). We say, that 2 and -2 are not in the domain of the function. The Domain of a function consists of all x-values that produce a y-value. These do not. The y values produced by the acceptable x-values are called the range of the function. The domain & range are a way of stating the scope of the function (values for which the function has meaning).

Since a functional value (y value in the range) depends upon an x value in the domain, then x is the independent variable while y is the dependent variable. In most applied problems in calculus, there is meaning to the functions used. For example, if we wanted to analyize the movement of a particle, we would consider the distance traveled as a function of time. In this case, we would use s for distance and t for time and the function would be described as s=f(t), where t is the independent variable. In business, if we were interested in profit as a function of items produced (x), we would consider a profit function P(x). So, depending on what we are analyizing the varibles may change.

Composition of Functions

This is a way of combining 2 or more functions in a special way. Think of a function as a mapping that takes a point in one set (pre-image) to a point in another set (image). (i.e., set A to image points in set B)(x to f(x))

The domain (accepted values) of the function f are the points in set A. A point in the domain of f, x, is taken from set A to its image under f to f(x) in set B.

The set of all images under f are points in set B. The set of these image points is the range of f.

There might be points in set B that are not image points under f, thus not in the range of f.

Let g be a function that takes image points in B to points in set C, f(x) to g[f(x)], where f(x) is a point in set B & g[f(x)] is it's image point in set C.

The function g takes points in the range of f (image points in set B) to their images in set C.

It is necessary that the image of points under f (these are the f(x)’s) are accepted by g (in the domain of g). If not, they will not have images in set C & the composition is not defined.

Let f(x) = -x²-1 and g(x) = √x. We see here that the composition is not defined since all points in the range of f are not accepted by g. That is, the range of f consists of negative numbers and the domain of g are just positive numbers. We will only consider compositions that are defined.

Let f(x) = x+3 and g(x) = x² . Find the function composition (fog)(x). By definition, (fog)(x) = f [g(x)] . This means, replace x in function f by g(x).
Therefore, (fog)(x)= f[g(x)] = f[x²]= x²+3.

If we reverse the composition and calculate (gof)(x) = g[f(x)] (this means, replace x in function g by f(x)), we get, (gof)(x) = g[f(x)] = g(x+3) = (x+3)². We notice that (fog)(x) does not equal (gof)(x).

Generally, function composition IS NOT a commutative operation.

The beginning of a function compositon does not always start with an operation on a single variable. For example, to do the composition (gof)(3x-1) using the above functions, we need to calculate g[f(3x-1)]=g[(3x-1)+3]=g(3x+2)=(3x+2)².

1-1 Functions

A function is 1-1, if for every image there is EXACTLY one pre-image. That is, for every value of f(x) there is exactly one value of x that corresponds to that value. (for each y-value there is exactly one x-value).

f(x) = x² is not 1-1. Since f(2)=4 and (-2)=4. So, we have a y-value of 4 with two x-values.

f(x) = x³ is 1-1. You cannot have 2 different numbers giving the same result when cubed.

Graphically, 1-1 functions satisfy the HORIZONTAL LINE TEST. That is, all horizontal lines that you can draw that intersect the graph, must do so in EXACTLY ONE POINT.

f(x)=x⁴ (not 1-1), g(x)=x³ (yes),   y=sqr(x)(yes),   y=2x+3 (yes)

1-1 functions are very special, they have inverses.

~How to find the inverse of a 1-1 function. Use the procedure below.

  1.   Replace function notation with y-notation. (i.e., replace f(x) by y).
  2.   Interchange x and y.
  3.   Solve the new equation for y (this is the inverse function).
  4.   Special notation is used for the inverse. f^(-1)(x) is the symbol for the inverse of f.

If f(x) = 3x - 2, find f^(-1)(x), the inverse of f. Note: f is 1-1, so f^(-1)(x) will exist

   1.   Replace f(x) by y:   y = 3x-2
   2.   Interchange x and y:   x=3y-2
   3.   Solve for y:    x+2=3y gives (x + 2)/3 = y   , so, f^(-1)(x) = (x + 2)/3


If f(x) = x³ , find f^(-1)(x). Note: f is 1-1, so f^-1 will exist.

          1.   Replace f(x) by y:   y = x³   .
          2.   Interchange x and y: x = y³
          3.   Solve for y:   y = x^(1/3) (cube root of x)

Inverses are very useful for finding pre-images if their images are known. (i.e., finding x, given f(x) or finding x, given y.)

Visualizing the inverse of a function

If we graph a 1-1 function and its inverse on the same set of axes, we will notice that they are reflections of each other about the line y = x. This reflection can easily be performed by folding a piece of paper.

The line y = x is a line with slope of one passing through the origin. To get the sketch of the inverse with a piece of paper, sketch the curve for which the inverse function is desired. Make sure it is a 1-1 function or the inverse will not exist. Also, do not label the axes. Then, you must make a crease along this line. Then fold the top part of the crease down and bottom part of crease back and up. Do not rotate the paper. You will then be looking at the back of the paper at the graph of the inverse function. Hold it up to a light source for a better view.