Finding the domain of a function can be as easy as finding the perfect life partner! Or as difficult as finding the perfect life partner.
Haha! Just kidding!
The Domain of a function has nothing to do with your life partner. Nor it is easy to find one. Sorry for the distraction! 😛
What actually is the Domain of a function? Let’s first understand its meaning.
In the next section, I’ll use a simple and my favorite analogy while explaining anything related to functions.
Meaning of the Domain of a Function
Let’s say you have a weighing machine that can tell you the weights of people ranging from 0 kg to 100 kg. And if a person stands on that machine whose weight is, let’s say, 123 kg, then the machine will get damaged; because it can’t withstand anything or anyone weighing more than 100 kg.
And if you keep something on the machine weighing less than 0 kg, then it will not show any result.
So, no result if the weight is below 0 kg and machine gets damaged if the weight is more than 100 kg. So, THIS interval of numbers (from 0 to 100) is actually nothing but the domain of our machine, if you consider the machine to be a function. That means you can only have a person from 0 to 100 kg on the machine.
Recall – A function is nothing but a machine that receives some inputs and gives you outputs on the basis of what the function is supposed to do.
So the domain can be defined as the set of those values which are allowed to be entered into the function (machine) without making it undefined (or damaging the machine).
Finding the Domain of a Function
Let’s consider a simple function, [latex] f(x) = \frac{1}{x} [/latex]
In this function, we can’t take x=0 as the function will become undefined, as division with zero is not possible. So the domain of this function becomes all real numbers except 0. Or [latex] R ~ {0} [/latex].
Now, we can summarize the above learning to find out the domain of rational functions, polynomial and square root functions.
Finding Domain of Rational Functions
As we know that rational functions are the functions that can be expressed as the ratio of two polynomial functions,
say [latex] f(x) = \frac{p(x)}{q(x)} [/latex]; where q(x) can’t be zero.
So, the domain of such a rational function will be all ‘x’ values at which the denominator ‘q(x)’ does not become zero.
Example: Consider the rational function [latex] h(x) = \frac{(x^2 – 9)}{(x – 3)} [/latex]. To find the domain, we need to ensure that the denominator, x – 3, is not equal to zero. Thus, the domain of h(x) is {x | x ≠ 3}.
Do one more problem to find out the domain of the rational functions by watching this video –
Finding Domain of Polynomial Functions
Polynomial functions are among the most common types of functions. They have the general form [latex] f(x) = a_n * x^n + a_(n-1) * x^(n-1) + … + a_1 * x + a_0 [/latex], where “[latex] a_n [/latex]” to “[latex] a_0 [/latex]” are constants, and “n” is a positive integer representing the degree of the polynomial.
Example: Let’s consider the polynomial function [latex] f(x) = 2x^3 – 5x^2 + 3x – 7 [/latex]. The domain, in this case, includes all real numbers since there are no restrictions on the variable “x.”
Square Root Functions
Square root functions have the general form [latex] f(x) = sqrt{(ax + b)} [/latex]. To ensure the expression inside the radical is non-negative (for square root), certain conditions must be met.
Example: Let’s examine the function [latex] f(x) = sqrt{(3x + 2)} [/latex]. To ensure the expression under the square root is non-negative, 3x + 2 ≥ 0. Solving for “x,” we find x ≥ -2/3. Therefore, the domain of f(x) is {x | x ≥ -2/3}.
Watch this video to understand it better –
Domain of a Function from its Graph
Finding the domain of a function from its graph involves identifying all the possible input values for which the function is defined and has a valid output. This is achieved by examining the x-values over the entire graph. Here’s a step-by-step guide on how to find the domain of a function from its graph:
Step 1: Identify the X-axis Range
Look at the x-axis of the graph to determine the range of x-values that are visible in the graph. The domain will generally be a subset of this range, but there may be exceptions where we exclude certain x-values.
Step 2: Note Any Vertical Asymptotes or Holes
Vertical asymptotes and holes in the graph can indicate excluded x-values from the domain. Vertical asymptotes are vertical lines where the function approaches infinity or negative infinity, indicating that the function is undefined at those points. Holes are points on the graph where there is an open circle, indicating that the function’s value does not exist at that specific x-value.
Step 3: Account for Any Discontinuities
Discontinuities in the graph can also signify excluded x-values from the domain. Discontinuities occur where the function has breaks or jumps and may be due to removable or non-removable singularities.
Step 4: Combine the Information
Based on the steps above, identify all the x-values that we need to exclude from the range determined in Step 1. The domain of the function will be all the x-values that remain after accounting for vertical asymptotes, holes, and discontinuities.
Example:
Let’s work through an example to find the domain of a function from its graph:
Consider the graph of the function f(x) shown below:
Step 1: Identify the X-axis range from the graph, we can see that the X-axis covers a range from -7 to 6.
Step 2: Note any Vertical Asymptotes or Holes. There are 2 holes in the graph, at x = -7 and x = -2 where there is an open circle. But at x = -2, the function is having a closed circle as well. So x can be -2.
Step 3: Account for any Discontinuities. The graph does not show any obvious discontinuities.
Step 4: Combine the information gathered, the excluded x-values from the domain are x = -7.
Therefore, the domain of the function f(x) from its graph is (-7, to 6].
Watch this video for ste-by-step explanation of the above problem –
