# How To Find Domain And Range Graphically – Step By Step Guide !!

We have seen all the different methods to find domain and range of a function in the last post of this series . Now it’s time to learn the smart way to find the domain and range of any function graphically .

In order to find domain and range of a graph , you have to learn the basics of plotting graph of any basic maths function . In this post, we’ll explore all the steps required to obtain domain and range of any given function quickly using a graph .

Initially, we’ll discuss the most common occurring standard real functions in the study of calculus . Also now let’s see the standard procedure that you need to follow in any problem involving graphs .

**How to Find Domain And Range Of A Graph ?**

Steps to be followed to find domain and range of a function graphically :

**Plot the graph of f(x) i.e. y = f(x) , for this you need to have the knowledge of graphs of basic maths functions****In any graph, we can have Domain as all the x – coordinate values (along x-axis) of the graph****And Range is all y – coordinate values (along y-axis) of the graph****Lastly we need to include/exclude the endpoints in the interval carefully by looking at the graph (for which f(x) is a valid function).**

Now the above set of steps is valid for any function involving graphs . If you already have a graph handy with you , then just go ahead and solve for domain and range (following step 2 onwards from above)

It’s time to now dig deep into each of the common functions and see how the range and domain look like indeed .

**How To Find Domain And Range Graphically ?**

As seen above, the first step is to plot the **graph of f(x)** . Here the foremost point is to know how a graph looks like for any given real-valued function. Now we gonna plot the graph of all commonly used functions and will see the domain and range in each specific case .

Always remember that in order to plot any random graph , you need to plot the coordinates in the cartesian plane that comprises four quadrants. In each quadrant, the value of the x-axis and y-axis will differ (either negative / positive) .

In order to make the plotting of f(x) simpler, you can just start plotting the x and y coordinates (of few **test points**) on the number line initially (to get the rough idea of the shape of the graph . ).

So if we have the point **(x,f(x))** , then start with the point (0,f(0)) , (1,f(1)) , and finally find the roots of f(x) . Plot the same on graph . So without wasting any further time let’s understand the graph of commonly used functions for your ready reference.

**Domain And Range Of Constant Function Graph !!**

Let’s start with the simple function first . So a constant function is basically a real number , which can be given as :

**f(x) = k …. for all x ∈ R**

Here f(x) is called a **constant function** . Graphically this can be represented as :

Clearly the graph will be a straight line running parallel to the x-axis (above/below the x-axis based upon the sign of ‘k’). If k = 0 , it will then coincident to x-axis .

For finding the domain of any graph we’ll start moving from the **left to right direction** (and see the starting and end of the graph along the x-axis) . Based upon this we’ll declare the domain of the graph .

For finding the range of the graph we’ll start moving from the **Downward towards the Upward direction** (and see the starting and end of the graph along the y-axis). Based upon this we’ll declare the range of the graph .

Since a constant function has no endpoint and it runs towards infinity. Based upon these two conditions, we can easily say that the **domain of f(x)** is **R** (all real values on the number line)

Also, the **range of f(x)** is also **R** (all real values i.e. -∞ to ∞ on the number line)

Hopefully, you have understood the process of finding domain and range from graph . Before we move over to the next function , let’s recap the definition of domain and range of a function .

**Domain** basically means , all the values of ‘x’ that the function f(x) can accept (possible input values of ‘x’) and **Range** is defined as the value of the function that will be obtained on putting the value of ‘x’ . (all output values of f(x))

**Domain And Range Of An Identity Function Graph !!**

Now it’s time to understand the Identity function . So an identity function is basically a function that associates each real number to itself , this can be given as :

**I(x) = x** …. for all x ∈ R

Here I(x) is called an **Identity function** . Graphically this can be represented as :

Clearly the graph will be a straight line running through the origin (and inclined at 45° with the x-axis).

For finding the domain of any graph we’ll start moving from the **left to right direction** (and see the starting and end of the graph along the x-axis) . Based upon this we’ll declare the domain of the graph .

For finding the range of the graph we’ll start moving from the **Downward towards the Upward direction** (and see the starting and end of the graph along the y-axis). Based upon this we’ll declare the range of the graph .

Since an identity function has no endpoint and it runs towards infinity. Based upon these two conditions, we can easily say that the **domain of f(x)** is **R** (all real values on the number line)

Also, the **range of f(x)** is also **R** (all real values i.e. -∞ to ∞ on the number line)

**Domain And Range Of Modulus Function Graph !!**

Now it’s time to understand the Modulus function . So a Modulus function is basically a function that can be given as :

Then **f(x) = |x| = x …. when x ≥ 0 ;**

Also , **f(x) = |x| = -x …. when x ≤ 0 ;**

Here f(x) is called a **Modulus function** . Graphically this can be represented as :

Clearly the graph will be a straight line running through the origin (and inclined at 45° with the x-axis) and then again a straight line running through the origin (and inclined at 45° with the x-axis) on the other side . (a mirror image indeed)

For finding the domain of any graph we’ll start moving from the **left to right direction** (and see the starting and end of the graph along the x-axis) . Based upon this we’ll declare the domain of the graph .

For finding the range of the graph we’ll start moving from the **Downward towards the Upward direction** (and see the starting and end of the graph along the y-axis). Based upon this we’ll declare the range of the graph .

Since an identity function has no endpoint and it runs towards infinity. Based upon these two conditions, we can easily say that the **domain of f(x)** is **R** (all real values on the number line)

Also, the **range of f(x)** is all positive values of **R** (all real values i.e. -∞ to ∞ on the number line)

Range of **modulus** function = **positive values of R i.e {x ∈ R : x ≥ 0}**

**Domain And Range Of Greatest Integer Function Using Graph !!**

For any real number ‘x’ , we use the symbol [x] to denote the greatest integer function (GIF). Further we can define a GIF function as the greatest integer less than or equal to ‘x’ i.e.

The function f : R → R defined by

**f(x) = [x] for all x ∈ R**

For ex : [1.76] = 1 , [7.45] = 7 , [-6.87] = -7 etc

GIF is also called as the **Step function** . Also if we plot the graph of GIF , clearly we can say that the domain of f(x) is set of all R (all real numbers on the number line) . The range can be computed graphically as set of all integers i.e. ‘Z’ .

The range is set of all **integer values** .

**Domain And Range Of Smallest Integer Function Using Graph !!**

For any real number ‘x’ , we use the symbol [x] to denote the greatest integer function (GIF). Further we can define a GIF function as the smallest integer greater than or equal to ‘x’ i.e.

The function f : R → R defined by

**f(x) = [x] for all x ∈ R**

For ex : [1.76] = 2 , [7.45] = 8 , [-6.87] = -6 etc

GIF is also called as the **Ceiling function** . Also if we plot the graph of GIF , clearly we can say that the domain of f(x) is set of all R (all real numbers on the number line) . The range can be computed graphically as the set of all integers i.e. Z

The range is set of all **integer values** .

**Domain And Range Of Signum Function Using Graph !!**

Signum function is one of the simplest functions having a well defined constant value of 1 . The function f can be defined as the below predefined conditions :

i.e. **f(x) = |x|/x x ≠ 0 ** ; ….. (the value of f(x) is 1 for x > 0 and -1 for x < 0)

And , f(x) = 0 , x = 0 ;

If we plot these conditions in graph , we can clearly see the domain and range values . For domain , just start moving from the left side towards the right side .

So now , **Domain of f(x) = R** (all values on the number line) ;

And **Range of f(x) = {-1 ,0 , 1} **

**Domain And Range Of An Exponential Function Using Graph !!**

Now for an exponential function a^x , we can define this as : if a is a positive real number other than unity , then a function that associates each x ∈ R to a^x is called an **exponential function** . It can be given as :

**f(x) = a^x** where a > 0 and a ≠ 1 is called the **exponential function** .

Now **domain of f(x) is given as R** and the range of f(x) is set of all positive values of R excluding ‘0’ i.e. **range = (0,∞) **as it attains only positive values **. **

Also , **f(x) = a^x** value is different for different values of ‘x’ and ‘a’ , but the values for domain and range will remain the same for all the cases .

**Domain And Range Of Logarithmic Function Using Graph !!**

A logarithmic function can be defined by :

**f(x) = for x > 0 ** ;

Also just keep in mind that the logarithmic and the exponential functions are inverse of each other . Now let’s plot the graph of the logarithmic function :

From the graph , it is pretty much evident that the domain of f(x) is set of all positive values i.e. (0,∞) and range of a logarithmic function is R (set of all real numbers)

**Note** : Always for determining the domain start moving from the left side towards the right direction , and for the range of function just move from the down to up direction .

**Domain And Range Of A Square Root Function Using Graph !!**

A square root function is defined as :

**f(x) = √x ** and f(x) is valid for only the positive values of ‘x’

Clearly the domain of this square root function is all positive values of ‘x’ i.e. [0,∞) and range of f(x) is also [0,∞) .

Now let’s see the graph of a square root function :

**Note** : Also the graph of a reciprocal function is given as :

**f(x) = 1/x ** , clearly domain is all R except ‘0’ and range is also R – {0}

**Domain And Range Of A Square Function Using Graph !!**

A square function will always result in a positive value , but it can accept any values in R . The square function is given by :

**f(x) = x² ** , is a square function .

Let’s plot the graph of f(x) as given below :

Clearly the domain of f(x) is R i.e. (-∞ , ∞) and the range of f(x) is [0,∞) , the graph of a square function is a parabola .

Now let’s try to find the domain and range of functions other than the standard function graphically . Always just keep in mind that in order to find the domain and range of a graph , you have to follow the below steps :

1. For finding domain graphically, start moving from the left side towards the right direction along the x-axis .

2. Now for finding the range of a graph , just move from the downwards direction to the upwards direction along the y-axis ,

The above two points will be clear from solving some examples to find domain and range graphically .

Let’s try to find the domain and range of a graph given below :

Clearly if we start moving from left to right direction , we’ll first encounter the graph of f(x) at x = -4 , then between the interval x = -2 to x = 1 , there is no graph present . Then again the graph of f(x) starts at x = 1 . Thus from all this , we can conclude that :

Domain of f(x) graphically is =** [-4 , -2] ∪ [1 , ∞) ;**

Now for range , we need to start moving from the downwards to upwards direction . Thus we’ll first encounter the graph at x = 0 , and also the graph will continue till infinity . So

Range of f(x) graphically is = **[0 , ∞)**

**How To Find Domain And Range Of A Function Graphically ?**

If you are given a function , the first step is to plot the **graph of f(x)** . Then from the graph, you just need to inspect the start and end points along the x-axis and y-axis for commenting on the domain and range value .

For eg : If **f(x) = 2x + 5 **

Clearly the x – coordinate value (at which f(x) is ‘0’) for this function is (-5/2) respectively . So with this knowledge, we can easily plot the graph as shown below :

Now if we start moving from the left to right direction , the domain ranges in the interval i.e. (-∞,∞) . And for the range value , we’ll just move from the downwards to up direction along the y-axis and clearly the range is also same i.e. R

**How To Find Domain And Range Of A Quadratic Function Graphically ?**

Now this is interesting. Firstly you have to plot the graph of the given quadratic function i.e. a parabola shaped plot precisely . Also, the opening of the parabola will depend on the coefficient of x² in the quadratic equation .

Let say **f(x) = ax² + bx + c **

Now the above equation is the standard quadratic equation , we can have different shaped parabola lying anywhere in the cartesian plane depending upon the roots and sign of the coefficient of ‘x²’

If **a > 0 **, then the graph of f(x) will be :

If **a < 0 ** , then the graph of f(x) will be :

Also remember that the **domain** of any quadratic equation will always be **R i.e. (-∞,∞) ** , which means that it can take any value for ‘x’ , and for the **range** we need to care for the start and endpoint of the quadratic equation along the y-axis . Let’s try an example now to make this concept clear :

Let’s try to find the domain and range of the function graphically :

**f(x) = x² + 5x + 6 **

Now first we need to plot the graph , for this let’s find out the roots :

by using the **middle term split up method** , we can write the above function as follows :

f(x) = x² + 5x + 6 = (x+2)(x+3) = 0 ;

The roots of the above equ is the point where f(x) = 0 ;

Hence the roots will be given as **x = -2 , -3 ;**

Now plotting the graph for the same :

**f(x) = x² + 5x + 6** ; the graph will look like as follows :

Clearly the domain is **R i.e. (-∞,∞) …..**(since there is no endpoint along the x-axis if we go from left to right)

Now range will be given as the points along the y-axis :

As seen in the graph , there is no endpoint of the graph in the upward direction along the y-axis. Now the start point of the graph is given as ‘**vertex**‘

where , the **axis of symmetry** = **midpoint** of both the roots on the number line

Also , for the y-coordinate of the **Vertex of f(x)** = Put the value of ‘**x**‘ if equ of f(x) …… (x is the axis of symmetry)

Now the graph will cut the y-axis at some point where , x = 0

**f(x) = x² + 5x + 6 ;**

On putting x =0 , f(x) = 6 ;

Now from the graph it is clear that the range of f(x) is given as [y coordinate of vertex,∞)

**Shortcut tip :** the vertex for any quadratic function can be given as the formula :

**Vertex = (-b/2a , -D/4a)** , where **D = b² – 4ac for the equation f(x) = ax² + bx + c ; **

On applying the formula , in the equation : f(x) = x² + 5x + 6 ;

a = 1 , b = 5 and c = 6 , Also D = 1

**Vertex of f(x) = (-5/2 , -1/4) **

The **range of f(x) is [-1/4 , ∞) **

I hope you really liked this post regarding the method to solve for the domain and range of a graph quickly . If you have any doubts, then do let me know the comments section. Stay tuned for more interesting stuff in this series .

Aric is a tech enthusiast , who love to write about the tech related products and ‘How To’ blogs . IT Engineer by profession , right now working in the Automation field in a Software product company . The other hobbies includes singing , trekking and writing blogs .