Calculus (Sixth edition) Chapter (1)--Function and Model

1.4 ways to represent a function

Each of the these is described a rule whereby, given a number x, and another number y is assigned. In each case we say Y (second number) is a function of x (first number).

Independent variable (dependent variable): a symbol that represents an arbitrary number in the domain of a functionƒ.

Dependent variable (independent variable): a symbol that represents a number in the range ofƒ.

If ƒ (x) is used as a machine, then the raw material is the input domain or x, the product produced by the machine is range, that is ƒ (x). For each x value, the function value ƒ (x) has a unique value corresponding to it.

Odd functions and even function

Even function (even function): If a functionƒsatisfiesƒ (-X) =ƒ (x) for every number x on its domain.

The even function is axisymmetric along Y.

Odd function (odd functions): ifƒsatisfiesƒ (-X) =-ƒ (x).

The odd function is axisymmetric along X.

The method of judging odd and even functions: replace x with X. To see what the result is.

Zang and subtraction functions

Note: The increment or decrement of a function is for a certain interval.

2. Function transformation

Horizontal and Vertical transformations:

Take y=ƒ (x) as an example:

y=ƒ (x) +c, the function moves up the C distance;

y=ƒ (x) –C, the function moves downward by a C distance;

Y=ƒ (x+c), the function moves to the left of the C distance;

Y=ƒ (x–c), the function moves the C distance to the right.

Stretching, shrinking, and reflection transformations:

Take y=ƒ (x) as an example:

y=cƒ (x), the function stretches the C-fold in the vertical direction;

y= (1/C) ƒ (x), the function shrinks the C-fold size in the vertical direction;

Y=ƒ (Cx), the function shrinks in the horizontal direction c times the size;

Y=ƒ (X/C), the function stretches the C-fold size in the horizontal direction;

Y=–ƒ (x), the function rotates 180° along the x-axis;

Y=ƒ (–x), the function rotates 180 ° along the y axis.

Compound function

Arithmetic of the function:

(Ƒ+G) (x) =ƒ (x) + g (x)

(Ƒ–G) (x) =ƒ (x) –g (x)

(ƒg) (x) =ƒ (x) *g (x)

(ƒ/g) (x) =ƒ (x)/g (x)

(Ƒ°G) (x) =ƒ (g (x))

Calculus (Sixth edition) Chapter (1)--Function and Model