Functions and Inverse Functions
Let us recall the definition of a function. If we have two sets A and B, a function from set A to set B is a correspondence between elements of A and elements of B such that
1. Each element of A corresponds to some element of B, and
2. No element of A corresponds to more than one element of B.
If the element x in set A corresponds to the element y in B, we write y = f(x), where f is the symbol for the function itself.
Example 73 Let us take A as the set of all real numbers, and B as another copy of the set of real numbers. If x is an element of A, then we can make it correspond to an element yin B by taking corresponds to some element y in B, (since any number can be squared), and no element x in A corresponds to more than one element yin B (since we get a unique answer when we square a number).
Our definition of a function is not very democratic. For every element of A, we must produce exactly one element of B. But if we have an element of B, we cannot tell if there is an element in A to which it corresponds.
An element of B may correspond to no element of A, to one element of A, or to more than one element of A .
In older texts, this undemocratic situation was described by calling x the independent variable and y the dependent variable.
Example 74 In Example 73, if we were given a number x in A, we are obliged to supply an answer to the question: what number y in B corresponds to A? For example, if x = 3, then we can answer that y = 9, and if x = -3, we can answer y = 9 again. This is allowed, under our definition of a function. The only restriction is that our answer must be a number in set B.
But if we choose an element y in set B, we are not obliged to answer the question: what number in A corresponds to it? Certainly, if we chose y = 9, we could answer x = 3. But we could just as well answer x = -3, and so our answer would not be unique. Worse, if we chose y = -1, we have no answer at all. There is no real number whose square is -1.
That is, if y is a function of x, it may not be the case that x is also a function of y. However, in some cases, we can improve the situation.
since the square of a real number cannot be negative. But now, if we take a number y in B, we can always answer the question: What number x in A corresponds toy? For example, if y = 9, we can answer that x = 3.
We are not embarrassed by the possibility of a second answer, since -3 is not in our (new) set A. Nor are we embarrassed by the lack of any answer.
Negative numbers, which are not squares of real numbers, do not exist in our new set B.
In general, we can take a function y = f (x ), try to start with a value of y, and get the corresponding value of x. If this is possible- if x is a function of y as well- then this new function is called the inverse function for f(x).
When does a function have an inverse function? This is an important question. We will not give a general answer here. We will, however, observe that if A and Bare intervals on the real line, then y = f(x), defined on these intervals, has an inverse if and only if it is monotone (steadily increasing or steadily decreasing). The first two graphs below show functions that are monotone, and have inverses. The last three graphs show functions.
that have no inverse on the sets A and B.
Arcsin: The inverse function to sin
Example 76 The equation y = sin x defines a function from the set A of real numbers to the set B of real numbers. Does it have an inverse function?
Again, the answer is no, and for the same two reasons as in Example 7 5. For some values of y in B, such as y = 5, there are no values of x such that
Example 82 Draw the graph of the function y = arcsin(sinx). Solution: We will soon see that this is not the same as the previous example(!). Again, we begin by deciding on the domain of the function.
We can take the sine of any real number x. Since the resulting value is in the interval from -1 to 1, we can then take the arcsine of this value. Hence the function y = arcsin(sinx) is defined for any real number x. The possible
Graphing inverse functions
How is the graph of a function related to the graph of its inverse function?
We can read the values of the function from the graph. For example, the diagram shows that f(2) = 4, since the x-value 2 corresponds to the y-value 4 on the graph.
contains all the information we need to find values of the inverse function.
We just choose our first number on the y-axis, and use the graph to get the corresponding number on the x-axis. For example, if we want g(4), we find the number 4 on they-axis, and use the graph to find the corresponding
number (which is 2) on the x-axis.
However, many people are more comfortable using the letter x to denote the number in set A for which the function is making an assignment, and the letter y for the number in set B to which x is assigned. There are two ways to accommodate this need. We can simply relabel the axes of the original graph:
But many people prefer the x-axis to appear horizontal, and the y-axis to appear vertical, on the page. We can accommodate them by reflecting the graph around a diagonal line:
This graph contains the same information as the others, but in a more conventional form.
Here are graphs of the sine function, and its inverse, the arcsine function. The graph of the inverse function is given in the conventional position.
Note that the domains are restricted as we discussed above.
And here are graphs of y = arccos x and y = arctan x:
The graph of y = arctan x shows clearly how the function maps the entire real line onto a finite interval.
We must often solve trigonometric equations: equations in which trigonometric functions of the unknown quantity appear. We can often use the following method to solve these:
1. Reduce them to the form sin a =a, cos x =a, or tanx =a;
2. Locate the solutions to these simple equations between 0 and 2rr;
3. Use the periods of the functions sinx, cosx, and tanx to find all the solutions.
We start with a simple example.
Example 86 Solve the equation sinx = 1/2.
This means that we must find all the values of x for which sin x = 1/2.
We will describe two ways of finding these values. Our first method uses a circle, and our second uses a graph of the function y = sin x.
Solution 1: We first use a unit circle, centered at the origin. As a first step, we find two particular answers. We recall that sin n /6 = 1/2.
Let us illustrate this on our circle. We draw an angle of n/6, and find the line segment which is equal to 1/2:
This is our first answer.
But if we draw a horizontal line across the circle, we find another angle whose sine is 1/2:
A more general trigonometric equation
Take some acute angle a. We wish to solve the equation sin x = sin a. One
solution is immediate: x = a.
More complicated trigonometric equations
Altogether, there are four sequences of solutions:
In conclusion, we note that we have already shown (Ch. 7, Appendix I.2; p. 159), that any trigonometric identity can be reduced to an algebraic identity. The same is true for trigonometric equations. However, the algebraic equation that results is often more difficult than the same equation in trigonometric form.
The Miracles Revealed
In Chapter 5 we discussed two small miracles:
The Miracle of the Tangent
If we draw a tangent to the curve y = sin x at the point x = a, then the distance between d, the point of intersection of this tangent with the x-axis, and the point (a, 0) is I tan a I.
The Miracle of the Arch
The area under one arch of the curve y = sin x is 2.
We now return to these results and furnish their proofs. Each draws on techniques that are standard in the study of the calculus. In particular, each uses the fact that the quotient sinh I h approaches 1 as h gets close to 0.
We showed why this is true on Chapter 5, p. 118. A more rigorous proof would involve the notion of limit, which is the fundamental notion of the calculus. In this section, we give a sketch of a proof for each miracle that parallels the more formal approach used in a course on calculus.
Proof of The Miracle of the Tangent
The diagram shows a point P(a, sin a) on the curve y = sin x. It intersects the x -axis at point R. We will show that Q R = I tan a I, by writing an equation for line P R, then finding the coordinates of point R.
We can write the equation of a line using the coordinates of a point on the line and the line’s slope. The point will be P, with coordinates (a, sin a).
To get the slope of line P R, we use a technique from the calculus.
Instead of looking at tangent P R, we look at a secant to the curve y = sinx, which intersects the curve near point P. We take two points, A and B, one just to the left of P and one just to the right, at a small distance h along the x-axis:
Now we take smaller and smaller (positive) values of h, so that points A and B get closer together, and secant AB begins to resemble tangent P R.
The expression sinh/ h gets closer and closer to 1 ash approaches 0. And of course cos a does not change as h approaches 0. So the slope of secant AB, which is looking more and more like tangent P R, gets closer and closer to the value cos a. It is reasonable, then, to expect that the slope of P R is exactly cos a. (In calculus, this technique of finding the slope of a tangent to a curve will receive a full justification. It is related to the notion
of the derivative of a function.)
Now we can find the equation of line P R, through point P(a, sin a) and with slope cos a:
Then the length of QR is just |a- (a- tan a) |= | tan a |.