Function of y=ax+b/x (a, b is not equal to 0)

1. when a>0 and b> are at 0 o'clock. Y=ax+b/x and y=x+ 1/x have the same shape, and the form of paired symbols is symbolic function.

This function has the following features:

The (1) sign function is obtained by hyperbola rotation, so it approximates straight lines, intersections, vertices, etc.

(2) The symbolic function is an eternal odd function, which is centrosymmetric about the origin.

(3) The two asymptotes of the symbolic function are always the Y axis and the straight line Y = X. ..

(4) When a>0, b>0, this image is distributed between the acute angles of two progressive lines in the first and third quadrants. Because of its symmetry, only the scene in the first quadrant is discussed. By using the important inequality, we can know that the minimum value is twice that of the root sign ab. It is obtained when x= root number b/a, so it monotonically decreases at (0, root number b/a) and monotonically increases at (root number b/a, positive infinity).

therefore

Let k= radical sign (b/a), then,

Increase the interval: {x | x ≤- k }∨{ x | x≥k}

Negative interval: {x |-k ≤ x

0 }∩{ x | 0 & lt；

x≤k}

2

When a<0, b<0, if y=x+ 1/x, the image of this function is,

f(x)

Is odd function, and the domain is X ..

Not equal to 0, simply increase at (negative infinity, 0), (0, positive infinity), the range is R, the asymptote is Y axis, and the straight line Y = X. Other situations of A and B can be obtained through the transformation of 4 and 5.

In short, as a symbolic function, it is very easy to remember that it is a hyperbola, so you can make an asymptote and find a special point to make the whole image.

As for how to judge the monotonicity of symbolic function, it can be proved by definition or judged by derivative, the latter is simpler.

3. Other attributes:

Asymptote y=ax

The size is related to the size of its opening.

The greater a, the greater the slope, the smaller the angle between y=ax and y axis, and the smaller the opening, and vice versa. When a equals 0, it becomes an inverse proportional function.