The best example for the domain and range of a function is the coffee vending machine. If you put a certain amount of money into the machine it will give you the coffee. Similarly, in **domain and range**, the domain of a function is the values that you put as input and the range of a function is the values that you get as output. Ex: Example: y = 2x+3, Let the x values be 2, 3, 4. So the resultant y values are 7, 9, 11. Here the values of y are dependent on x. Hence x is an independent variable and y is a dependent variable. The values of independent variables belong to the domain of the function and the values of dependent variables belong to the range of the function. The values of the domain are called preimages and the values of range are called the images.

**Definition of Calculus**

Calculus represents the rate of change of dependent and independent variables. Both differential and integral calculus serve as a base for the higher branch of Mathematics known as “Analysis”, dealing with the effects of a small change in dependent variable, as it leads to zero, on the function. Precalculus includes the study of trigonometry and algebra. For a better understanding of calculus, one must learn precalculus.

**Domain and Range of Various functions**

### Domain and Range of Exponential Functions

An exponential function is a function that has an independent variable as an exponent. The general form of exponential functions is = *f* (*x*) = *a*x, where *a* > 0, *a*≠1, and *x* is any real number. if a is negative, the function is undefined for -1 < *x* < 1. The positive values of ‘a’ allow the function to have a domain of all real numbers. Here *a* is called the base of the exponential function. The domain of exponential functions = R and the Range = (0,).

### Domain and Range of Trigonometric Functions of calculus

As we know, trigonometry belongs to the category of Precalculus. Understanding this concept is a must before studying **calculus**. Here we learn about finding the domain and range of trigonometric functions.

Let us consider a trigonometric identity:

sin2 + cos2 = 1

From the given identity,

cos2 = 1- sin2

cos = (1-sin2)

Since the cosine function is defined for real numbers the value inside the root is greater than 0

Therefore, 1 – sin2 0 1 sin2 and hence sin {-1, 1}

Thus we got the domain and range of sin function. Similarly, we can find the domain and range of other trigonometric functions also.

### Domain and Range of a Square Root Function

The function y = px + q is defined only for *x* ≥ – qp.The domain of the square root function is the set of all real numbers ≥ ‘- qp’. Since the square root results in a positive value always, the range of a square root function is the set of all positive real numbers. Hence, the domain of square root function = (-b/a,∞) and the range = [0,∞]

### Domain and Range of an Absolute Value Function

An absolute function y=|ax+b| is defined for all real numbers. So, the domain of the absolute value function is the set of all real numbers. The absolute value of a number is always a positive number. Thus, the range of the given absolute value function is y ∈ R | y ≥ 0.

Domain and range of a function can be determined by plotting a graph also.

Example: Plot a graph for the function y = 3x where the values of x are 4,8, and 12.

For this you have to first find the values of y for different values of x

y = 34 = 12 for x = 4, y = 38 = 24 for x = 8 and y = 312 = 36 for x = 12.

Plot the graph. Now the domain = 4,8,12 and Range = 12,24,36.

This method is suitable for finding the domain and range of quadratic equations and trigonometric functions. For more information log on to the Cuemath website.