Domain and Range of a Function

Domain and range are the most important parts of a function. They work just like a vending machine in this case. You can buy chips and soft drink cans with bill notes, but if you put in coins, the machine will reject them. Similarly, the domain represents the inputs available for a function, just like notes. Also, you cannot buy pizza from the machine no matter how much you pay. The range of a function works like that. It represents the possible outputs that can be obtained which, in the above example, are chips and soda. In this article, we will discuss the domain and range of a function.

What are Domain and Range?

The domain and range of a relation are defined as the sets of all the x-coordinates and y-coordinates of the ordered pairs, respectively. For instance, if the relation is,

R = {(1, 2), (2, 2), (3, 3), (4, 3)}, then:

Domain = the set of all x-coordinates = {1, 2, 3, 4}

Range = the set of all y-coordinates = {2, 3}

Methods Domain And Range of a Function Is Calculated?

Assume X = {1, 2, 3, 4, 5,}

f: X→ Y, and R = {(x,y): y = x+1}.

The domain is made up of the input values. As a result, domain = X = { 1, 2, 3, 4, 5}

Range = the function’s output values ={ 2, 3, 4, 5, 6} as well as the co-domain = Y = {2, 3, 4, 5, 6}

Let’s look at the domain and range of some special functions, taking various types of functions into account.

Solved Questions on Domain Range

Question: Find the domain of the following function:                       

(x2)/(x-4)(x-6)

Solution: It is critical to note that values in the domain of x should not make function undefined, so we must investigate the values which can cause it to be undefined or infinity.

Looking at the denominator, values 4 and 6 make the denominator 0 and thus make the function infinite, which is undesirable.

As a result, the values x=4 and x=6 cannot be used here.

R – {4,6} will be the domain.

Question: Find the domain values for which the functions Y = (2×2+2) and Z= (-4x) are equal.

Solution: Equating the two functions:

2 x2+ 2 =  – 4 x

2x2 + 4x + 2 = 0

2x2 + 2x+2x  + 2 = 0

2x (x + 1) + 2 (x+1)= 0

2(x + 1) (x + 1) = 0

x = -1

Therefore, the domain value is {-1}.

Question: Find the range. f(x) = √x – 1

Solution: Since the function is a square root, it can never produce negative results. As a result, the minimum value at x = 1 can only be 0. As we increase x, the maximum value can reach infinity.

Therefore, the function’s range is [0,∞].

Question: The domain of the function ƒ defined by f(x) = (1/x- |x|)

Given f(x)= 1/x-|x|

Solution: When choosing a domain set, two things must be considered.

The denominator is never zero. The term that is contained within the square root does not become negative. Let us expand on what is written inside the term within the square root.

In this case, we cannot use the value x≥ 0.

F is not defined for any x ∈ [0,∞]. As a result, the domain is[-∞,0]. 

Question: Plot the graph of the following function f(x) = 2×2 + 4x + 2. 

Solution: When the above equation is compared with general form ax2+bx+c. a is equal to 2, b is equal to 4, and c is equal to 2.

The parabola will open upwards as a > 0.

Value of x coordinate for vertex = -b/2a =-4/4

 = -1

Value of y coordinate for vertex = 2(-1)2 + 4(-1) + 2 = 0

As a result (-1,0) is the vertex of the parabola. Because the parabola opens upwards, this must be the function’s minimum value.

The point at which the graph intersects the y-axis is (0,2).

Question: Find the domain and range of the following function.

f(x) = | x – 16 |/(x – 16)

Solution: For values of x other than x = 16, f(x) assumes real values.

Domain = R – {16}, where R denotes the set of all real numbers.

If x is greater than or equal to zero, then |x| = x.

By using the rule mentioned above, we get

When x > 16, f(x) = (x – 16)/(x – 16) = 1.

When x <16 = (- (x – 16))/((x – 16)) = -1

Therefore Range ={- 1, 1}

Conclusion

A function’s domain and range are its components. The domain of a function is the set of all its input values, while the range is the function’s possible output. In this article, we have studied domain range, domain and range of a function, and the range of functions.