## Define Hyperbolic Function in Simple Way

• Sin Hyperbolic Function
• Cos Hyperbolic Function

### Simply Define sinh x hyperbolic function

Sinh x = \frac12\left(e^x-\;e^{-x}\right) is called hyperbolic sine function. Its domain and range are the set of all real numbers.

### Formula for Cosine hyperbolic function in Math with Detail

Cosh x = \frac12\left(e^x+\;e^{-x}\right) is called hyperbolic cosine function. its domain is the set of all real numbers and the range is the set of all numbers in the interval.

[1, +∞)

### Four Hyperbolic Functions and Formulas and Their Details

The remaining four hyperbolic functions are defined in the terms of the hyperbolic sine and the hyperbolic cosine function as follows:

### Four Hyperbolic Function Formulas are

The hyperbolic functions have the same properties that resemble to those of trigonometric functions.

## What is an Asymptote in Calculus

In mathematical terms, an asymptote is a curve that approaches infinity, asymptotes are lines that appear parallel to it.

### Types of Asymptotes in Calculus

There are three types of Asymptote in math.

### How to find Horizontal Asymptote of Exponential Function?

Horizontal Asymptote: A horizontal asymptote occurs when A curve approaches some constant value b as x approaches infinity (or *infinity)..

### How to find Vertical Asymptote of Exponential Function

Vertical Asymptote: A horizontal asymptote occurs when a function of X approaches some constant, c (from the left or right) then the curve continues to infinity (or −infinity).

### What is Oblique Asymptote?

Oblique Asymptote: A Oblique Asymptote occur when, as x goes to infinity (or −infinity) the curve then becomes a line y=mx+b

### Asymptote for a Curve Definition in Math

Definition: A straight line l is called an asymptote for a curve C if the distance between l and C approaches zero as the distance moved along l (from some fixed point on l ) tends to infinity.

The curve C can approach asymptote l as one moves along l in one direction, or in the opposite direction, or in both directions.

### How to Find Equation of Horizontal Asymptote?

Suppose the equation y\;=\;f(x) of C is such that y is real and y\rightarrow a as X\rightarrow\infty or X\rightarrow-\infty then y\;=\;a is a Horizontal asymptote. For, the distance between the curve and the straight line y\;=\;a is y\;-\;a and this approaches zero as X\rightarrow\infty or X\rightarrow-\infty.

### How to Find Equation of Vertical Asymptote?

If the equation of C is such that y is real and Y\rightarrow\infty or Y\rightarrow-\infty as x\rightarrow a from one side then the straight line x = a is a vertical asymptote.

To see this, observe that (1) x – a is the distance between the curve and the straight line and that this distance is supposed to approach zero (2) Y\rightarrow\infty or Y\rightarrow-\infty as x\rightarrow a , so that

\lim_{y\rightarrow\pm\infty}\;(x-a)\;=\;\lim_{x\rightarrow a}\;(x-a)\;=\;0

Thus, to locate vertical asymptotes we have to find a number a such that \lim_{x\rightarrow a}\;y\;=\;\infty\;or\;-\infty

Similarly, if y\rightarrow mx\;+\;c as x\rightarrow\infty or x\rightarrow-\infty then y\;=\;mx\;+\;c is an asymptotes ( which is neither vertical nor horizontal).

Thus, we inquire for \lim_{x\rightarrow\pm\infty}\;y is studying asymptotes.

### Find Horizontal and Vertical asymptote for Algebraic equations

For Algebraic equations, we can find horizontal and vertical asymptotes as follows:

For horizontal asymptotes we write the given equation in the form x\;=\;\frac{\psi(y)}{\theta(y)} and consider those values of y for which \theta(y)\;=\;0.

Similarly, to find a verticle asyptote, we write the given equation in the form y\;=\frac{f(x)}{g(x)} and consider those value of x for which g(x) = 0.

### Working Rule for Asymptotes Parallel to the Axes

In an equation of a curve, the coefficient of the highest power of x (respectively of y) equated to zero gives asymptotes (if any) parallel to the x-axis (respectively y-axis)

## Defined Graphs of Algebraic Functions

If f is a real-valued function of a real numbers, then the graph of f in the xy-plane is defined to be the graph of equation y = f(x). The graph of a function f is the set of point { (x, y)| y = f(x), x is in thedomain of f } in the cartesian plan for which (x, y) is an ordered pair of f.

The graph provides a visual technique for determining whether the set of points represents a function or not. if a vertical line intersects a graph in more than one point, it is not the graph of a function.

Explain is given in the figure.

### Best Method to Draw the graph

To draw the graph of y = f(x), we give arbitrary value of our choice to x and find the corresponding values of y. In this way we get ordered pairs \left(x_1\;,\;y_1\right) , \left(x_2\;,\;y_2\right) , \left(x_3\;,\;y_3\right) etc. These orders pairs represent points of the graph in the cartesian. We have these points and join them together to get the graph of the function.