10. Using Huntington's Tables, construct a table of values of x for x = 0.05, 0.10, 0.15, 0.20, 0.25, obtaining the values of the function to two decimal places. Construct the graph from x = 0 to x = 0.25 on as large a scale as possible.

11. Plot the graph of x3 4x. On the same axes sketch the graph of the inverse function, and state several of its properties by interpreting its graph. Can you find the inverse function?

NOTE. In finding the inverse of a function, it is necessary to solve an equation. At any stage of mathematical development, the solution of an equation may be impossible by means of functions already studied. Such an equation defines a new function, whose fundamental properties are determined by the equation.

Thus it is impossible for a student beginning algebra to solve the equation y2 = x for y. It is first necessary that he should become acquainted with the, to him, new function x1 == ±√x.

A person unacquainted with the solution of cubic equations cannot find the inverse of the cubic function in Exercise 11. But the theorem on the graphs of inverse functions enables us to get the graph and some of its properties, even though we do not know the function. point of view will be useful later in studying certain transcendental functions.

This

The inverse of a cubic or biquadratic function (page 39) is an algebraic function, but the inverse of a polynomial of higher than the fourth degree is usually transcendental.

12. What is the inverse of the function 1/x? What therefore can be said of the symmetry of the graph of the function?

13. Show that the graph of an equation is symmetrical with respect to the line y x if the equation is unchanged when x and y are interchanged. Plot the graph of xy - 2x - 2y = 0.

=

14. What is the form of an equation in x and y if it defines a function of x which is its own inverse?

15. Plot the graphs of x2 and

lowing facts are illustrated. (a)

on the same axes.
(√x)2 = x. (b) √x2

16. Plot the graphs of r3 and x on the same axes.

Show how the fol

= x.

Show how the fol

lowing facts are illustrated. (a) (Vx)3 = x. (b) Vx3 = x.

17. For what value of x does x2 increase at the same rate as x? For what value does x2 increase less rapidly? more rapidly? interpret the results graphically.

18. Find the value of x for which the tangent lines to the graphs of x2 and r3 are parallel. For what values of x is the graph of x3 "flatter" than that of x2? for what values is it steeper?

19. The horse power of a gas engine is sometimes determined by the equation H.P. d2n/2.5, where d is the diameter of a cylinder and n is the number of cylinders.

=

(a) Plot the graph for four-cylinder engines, taking d = 2.5, 3, 3.5, 4. On the same axes, plot the graphs for six-, eight- and twelve-cylinder engines. (b) Plot the graph for d 3, taking n = 4, 6, 8, 12. On the same axes, plot the graph if d = 2.5, or d

=

=

4.

40. Graph of x", n<0. Graphs of Reciprocal Functions. Consider first n = 1, the function x-1= 1/x. Let y = 1/x. Replacing x by x and y by - y, and changing the signs of both sides of the equation, we obtain the given equation. Hence the graph is symmetrical with respect to the origin, and the table of values need include only positive values

of x.

Since the function becomes infinite as x approaches zero, the y-axis, x = 0, is an asymptote. Solving y = 1/x for x we get x = 1/y, from which it follows that the x-axis, y = 0, is also an asymptote.

The figure shows the graph, which appears to be symmetrical with respect to the line

y

=

x

y

-3H

-2

1/x

=

1,

x, the bisector of the first and third quadrants. That this is the case is seen as follows: The equation y = 1/x, or xy is unchanged if x and y are interchanged. Hence if (x, y) is on the graph, so also is (y, x). But these points are symmetrical with respect to the line y = x (Section 39), and therefore the graph is also.

The graphs of x and 1/x illustrate the following properties of any two reciprocal variables:

versa.

I. Reciprocal variables have the same sign. For both graphs are above the x-axis, or both are below, for any value of x. II. If a variable increases, the reciprocal decreases, and vice For as x increases, that is, as the graph of x rises, the graph of 1/x falls; and as a decreases, the graph of 1/x rises. III. If the numerical value of a variable approaches and becomes unity, so also does that of the reciprocal. For the graphs intersect at the points (1, 1) and (− 1, − 1).

IV. If a variable approaches zero, its reciprocal becomes infinite, and vice versa. For both axes are asymptotes.

n = 2, the function x 2 = 1/x2. The function 1/2 is called the reciprocal of the function 22 in accordance with the DEFINITION. Two functions are said to be reciprocal if their product is unity.

The following properties of the graphs of reciprocal functions are proved by the like numbered facts above:

Ia. Corresponding parts of the graphs of reciprocal functions lie on the same side of the x-axis.

IIa. If the graph of a function rises, the graph of the reciprocal function falls, and vice versa.

IIIa. If the graph of a function approaches a point on either of the lines y = 1, the graph of the reciprocal function approaches the same point from the opposite side of the line. The points of intersection of the graphs lie on these lines.

IVa. If the graph of a function crosses, or is tangent to, the x-axis at the point (a, 0), the line x = a is an asymptote of the graph of the reciprocal function, and vice versa.

If the graph of a function has been plotted, these considerations enable us to sketch, roughly, the graph of the reciprocal function, without building a table of values.

The general form of the graph of 1/2 may be obtained from that of 22 as follows:

At the extreme left of the figure, the graph of x2 lies above the x-axis, and is falling. Then the graph of 1/2 lies above the x-axis, and is rising (by Ia and IIa). As x increases up to x the graph of x2 falls until it reaches the line y = 1, and hence the graph of 1/2 rises until it reaches this line

=

- 1,

(by IIIa). From these facts we can sketch the graph of 1/2 from the extreme left of the figure up to x 1.

=

As x increases from 1 to 0, the graph of x2 falls and be

comes tangent to the x-axis. Hence the graph of 1/2 rises indefinitely, and approaches the y-axis as an asymptote (by IIa and IVa).

As x increases from 0 to 1, the graph of x2 rises to the line

y = 1, and hence that of 1/22 falls to this line. As x increases from 1 on, the graph of x2 rises indefinitely, and that of 1/x2 continues to fall, but remains above the x-axis. The right hand half of the figure might also be obtained from the symmetry with respect to the y-axis.

Notice that the graph of 1/2 lies between the graph of 1/x and the x-axis to the right of x = 1, while between x = 0 and x = 1, the graph of 1/x lies between it and the y-axis.

41. Summary of Graph of x". The figure shows the typical form of the graph in the first quadrant for each of the cases n>1, 0<n<1, and n<0. This part of the graph always lies in the parts of the

The following remarks apply only to the part of the graph in the first quadrant.

n>1. The graph is tangent to the x-axis at the origin, and rises as it runs to the right. The larger the value of n, the flatter the graph is near the origin, and the steeper it is elsewhere.

0<n<1. The graph is tangent to the y-axis at the origin, The smaller the value of n the more rapidly it rises near the origin, and the less rapidly elsewhere.

and rises as it runs to the right.

The curves of these two groups are separated by the graph of yx (n = 1); the graphs for two reciprocal values of n are the graphs of inverse functions and are symmetrical to each other with respect to this line.

n<0. Both axes are asymptotes of the graph, which falls as it runs to the right. The smaller the numerical value of n, the closer the graph lies to the y-axis, between x = 0 and x = 1, and the farther it lies from the x-axis.

The curves of this last group are separated from those of the first two groups by the line y = 1 (n= 0), the graphs for two values of n which are numerically equal but differ in sign being the graphs of reciprocal functions.

For three values of n in any one of these three groups, the graph corresponding to the intermediate value of n always lies between the graphs for the other values of n.

The graph of ax" may be obtained from that of " by the theorem in Section 30.

EXERCISES

1. Find the reciprocal of each of the following functions, and sketch the graphs of both functions:

(d) x.

(e) x3.

2. Find the inverse and the reciprocal of each of the functions below, and sketch the graphs of the three functions on the same axes.

(a) x2.

3. On the same axes, plot the graph of the function indicated below for the given values of a.

(e) a/x2 for a = 1, 2, 4, 6.

4. On the same axes, using as large a scale as possible, construct the graph of x" for n - 2, and 4. What can be said of the form of the graph if n is an even negative integer?

=