§ 3.2. Composite Functions

Definition. A function of the form y=f(u), where u=(x), is called a composite function and denoted by у=f[(x)]; it is also the composition of the functions f and .

For example,

If y=u³–1 and u=lgx, then y=lg³x–1 is a composite function;

if y=a^u, u=t²and t=sinx, then is a composite function.

Representation of a composite function by a chain of elementary functions. To differentiate composite functions, it is important to decompose them into elementary functions.

For example, the composite function

1. y=ln⁴(tanx) can be represented as y=u⁴, u=lnz, z=tanx;

2. if , then y=u⁵, u=arcsinz, z=a^t -1, t=x²;

3. if y=ln⁷{cot(x³–1)⁵}, then y=u⁷, u=lnz, z=cott, t=⁵, =x³-1.

All functions in terms of which these composite functions are represented are elementary.

We also consider composite functions coinciding with different functions on different intervals of the domain.

For example,

y

y=2x

y=8

y=x²+1

0 1 4 x

Definition. If a function coincides with different elementary functions on certain intervals, then it is called a composite function.

Definition. A function х=(у) is called the inverse function for a function y=f(x) if the substitution of the former into the latter yields a true identity: y=f[(y)].

To find an inverse function, it is necessary to express x in terms of y from the equation y=f(x).

Example.

y=x²–1, x²=y+1, .

Remark. Sometimes, the variables in an inverse function are interchanged:

then , . Interchanging the variables we obtain y=.

Definition. A function y=f(x) is said to be periodic with period Т if

f(x+T)=f(x)

for all х from the domain.

Definition. A function is said to be implicit if its equation is given in the form

F(x;y)=0,

i.e., у is not explicitly expressed in terms of х.

Example. х²+у²-3=0;

x³siny–y⁵+16=0;

.

§ 3.3. Polar Coordinates

Take a point 0 and draw a ray from it. The point 0 is called a pole. The ray is called a polar axis. Choose a unit length. Take a point М in the plane and join it with the pole. The length ОМ= is called a polar radius. There are infinitely many points with the same polar radius. To determine a point uniquely, we must know the angle , which is called the polar angle. Any point in the plane is determined by two coordinates (;), which are called polar.

M(;).



0

There are two types of polar coordinates:

(1) strict polar coordinates, where 0,

(2) general polar coordinates, where both 0 and <0 are allowed.

Example. Construct the points , , and .

Remark. Points with negative polar radii are reflections of points with positive radii.

We shall consider generalized polar coordinates.

3.3.1. The relationship between the polar and Cartesian coordinate systems. Consider Cartesian rectangular coordinates. Take the polar coordinate systems such that the pole coincides with the origin, and the polar axis coincides with the positive direction of the x-axis. From ОМК we express the polar radius  and the angle  in terms of the rectangular coordinates х and у:

у М(;)

М(х;у)

ρ

у

 К

0 х х

Conversely, х and у are expressed from the triangle ОМК as

(*)

Formula (*) establishes the relationship between the polar coordinate system and Cartesian coordinate system.

Example. Construct the curve =аsin2 in generalized polar coordinates.

Let us compose a table of values of the polar angle and the radius.

	
0	0
	а
	0

This curve is called the four-leaf rose.

The reader is suggested to find the values of the polar angle and polar radius in the second, third, and forth parts of the plane.

The more the number of values, the better the picture.

Remark. The same function =аsin2 is represented as a two-leaf rose in some calculus text- and problem-books. (Why?).