Functions (trigonometry)

From Wikiversity

One of the most neglected topics in high school is the study of functions. In this lesson there are three rather lengthy chapters directly related to functions and several others that are indirectly related. There are two reasons for this: functions are important, and most calculus courses assume you know this topic almost perfectly, an unrealistic assumption. So let's get started at the beginning.

Function: Given a set D. To each element in D, we assign one and only one element.

Example 1

Does the picture here represent a function? The answer is yes. 1 goes into a, 2 goes into 3, 3 goes into 3, and 4 goes into pig. Each element in D is assigned one and only one element.

                                   D
                                   1    →    a
                                   2 →(to 3)
                                             3
                                   3 →(to 3)
                                   4    →    pig

The next example will show what is not a function. But let us talk a little more about this example. The set D is called the domain. We usually think about x values when we think about the domain. This is not necessarily true, but it is true in nearly all high school and college courses, so we will assume it.

There is a second set that arises. It is not part of the definition. However, it is always there. It is called the range. Notice that the domain and the range can contain the same thing (the number 3) or vastly different things (3 and pig). However, in math, we deal mostly with letters and numbers. The rule (the arrows) is called the map or mapping. 1 is mapped into a; 2 is mapped into 3; 3 is mapped into 3; and 4 is mapped into pig.

[edit] Functional Notation

The rule is usually given in a different form: f(1) = a (read "f of 1 equals a"); f(2) = 3; f(3) = 3; and f(4) = pig.

NOTE 1

When we think of the range, we will think of the y values, although again this is not necessarily true.

NOTE 2

We cannot always draw pictures of functions, and we will give more realistic examples after we give an example of something that is not a function.

Example 2

The picture here does not represent a function, since 1 is assigned D R two values, a and d. a

                                                                          1 →(to a and d) 
                                                                                            d
                                                                          2        →        e
                                                                          3 →(to f)
                                                                                            f
                                                                          4 →(to f)

Example 3

Let $y = f(x) = x^2+4x+7 \,$ . The possible values of $x\,$ form the domain.

Let us assume that there are only three possible values of $x\,$ , that is, $\mathcal{D} = \{1,-3,10\}\,$ .

Then, the function $f(x)\,$ converts these three values of $x\,$ into three different values of $y\,$ which form the set $\mathcal{R}$ . The set $\mathcal{R}$ is called the range of the function.

The adjacent diagram shows a couple of ways that you can express what a function does. You can try to come up with your own ideas on other ways to express the same thing.

Functions.

In this example, the three values that form the set $\mathcal{R}$ are found to be

$\begin{align} f(1) & = (1)^2+4(1)+7 = 12 \\ f(-3) & = (-3)^2+4(-3)+7 = 4 \\ f(10) & = (10)^2+4(10)+7 = 147 \end{align}$

Therefore the range is $\mathcal{R} \equiv \{4,12,147\}$ . If we were to draw the points $[x, f(x)]\,$ on a graph, we would have to graph (1,12), (-3,4) and (10,147).

NOTE

Instead of graphing points (x,y), we are graphing points (x,f(x)). For our purposes, the notation is different, but the meanings are the same.

Example 4

Let g(x) = x²-5x-9. D = {4,0,-3, $a 4$ ,x+h}. Find the elements in the range.

This is a pretty crazy example, but there are reasons to do it.

  g(4) = (4)²-5(4)-9 = -13 
                
  g(-3) = (-3)²-5(-3)-9 = 15       
        =  $a 8$ - $5 a 4$ -9

  g(0) = 0²-5(0)-9 = -9

  g( $a 4$ ) & = ( $a 4$ )²- $5 a 4$ -9

  g(x+h) = (x+h)²-5(x+h)-9                                                  Wherever there is an x,  
         = x²+2xh+h²-5x-5h-9                                                you replace it by x + h!

The range is {-13,-9,15, $a 8$ - $5 a 4$ -9,x² +2xh +h²- 5x -5x -9}.

Example 5

Find

$\frac{f(x+h)-f(x)}{h}$ .

$\frac{f(x+h)-f(x)}{h} = \frac{x+h} {x+h+5}-\frac{x}{x+5}/{h}$ (Add the fractions. Two tricks: a/b-c/d=(ad-bc)/bd;(e/f)/h=e/fh.)

$\frac{f(x+h)-f(x)}{h} = \frac{(x+h)(x+5)-x(x+h+5)}{(x+h+5)(x+5)h}$ (Multiply out the top; never multiply out the bottom.)

$\frac{f(x+h)-f(x)}{h} = \frac{5h}{(x+h+5)(x+5)h} = \frac{5}{(x+h+5)(x+5)}$ (Cancel the h's.)

This kind of problem occurs in almost every precalc book. What you should ask is why the heck it is here. I will tell you. This is very close to the topic you deal with in calculus. Here is a preview.

We have learned that the slope of a straight line is always the same. However, if we draw any curve and draw all its tangent lines, the slope changes. We would like to study this and algebraize it.

Given the point $P(x,f(x))\,$ . A little bit away from P is point Q. Its x value is x + h, where x + h is an x value a little bit away from x. If the first coordinate is x + h, the second coordinate is f(x + h). Draw PQ, PR (horizontal line), and QR (vertical line). On any horizontal line all y values are the same. P and R have the same y values. Q and R have the same x values.

Since Q and R have the same x values, the length of QR, which is the change in y or Δy, is $f(x + h) - f(x)\,$ . Since P and R have the same y values, the length of PR, which is the change in x or Δx, is $(x + h) - x - = h\,$ .

The slope of the secant line l₂ joining the points P and Q is

$\frac{\Delta y}{\Delta x} = \frac{f(x+h)-f(x)}{h}$

which is why we study this expression. But here's the conclusion. If we let h go to 0, graphically it means the point [x+h,f(x+h)] gets closer and closer to [x,f(x)]. If we do this process to the left of P as well as here to the right of P, and if they both approach the line l_1, then what we have calculated is the slope of the tangent line l₁ at the point [x,f(x)]!!!!! You have taken your first step into calculus!!!!!

1-1 function. 1-1 is a property we need occasionally.

DEFINITION

If f(a) = f(b), then a = b.

Examples 6 and 7

f(x) = 2x is 1-1, but g(x) = x² is not 1-1.

If f(a) = f(b), then 2a = 2b and a = b.

If g(a) = g(b), then a² = b². But a could equal b or -b; therefore, not 1-1.

How to tell a function by sight: Use the vertical line test. If, for every vertical line, each line hits the curve once and only once, then we have a function (for each x value, there is only one y value). If there is one vertical line that hits a curve twice, the curve is not a function.

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Functions (trigonometry)

From Wikiversity

[edit] Functional Notation

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