A simple way to identify the domain and range of functions is using graphs. But before that, you need a clear understanding of what domain and range mean. Domain means a set of potential values or x-values. These values are the input of a function, while the range is the possible output values or y-values of a function.
In graphs, domain and range are common values, but you need to identify them clearly. In this post, we have shared instructions on how to get the domain and range from the graph of a function.
Definitions of Domain on a Function
The domain contains a complete set of potential values of a function, and the function contains an independent variable. The domain contains all the possible x-values, which help the function to run properly.
The running function will then generate real y-values with the help of the x-values. When you are trying to find the domain, just remember that you cannot set the denominator (bottom) of a fraction to zero. And also, the number under a square root should be positive and not odd. For example:
Here the domain is x≥−4 because x should not be less than -4. You can bring out your calculator and try out numbers less than −4, such as -5 or -10. Or use numbers that are greater than -4, such as -2 or 8. The numbers that will give us an answer equal to -4 or will be greater than -4 are going to be the numbers that work. The number will make another positive number under the square root.
How Do You Find the Domain?
You can find the domain of an individual function by searching for the independent variable values or x-values. Generally, you can use these values when they are not 0 on the bottom of the fraction or negative value under the square root sign.
Definition of Range on a Function
The range of a function is a complete set of values related to the dependent variable or y in this context. The range refers to possible output values that we get after substituting all the possible input values or x.
How to Find the Range?
The range is the set of minimum y-values to maximum x-values available in a function. Whenever you find a function with its x and y values attached, see the positive numbers in the equation. You will have to substitute different x-values from y to find the minimum and maximum values. You will have to look for the minimum and maximum values for y.
For example:
Here are curves on or above the horizontal axis, so in this case, you will always get a positive value of y even if it is a zero. Here, the range is y ≥ 0. But since the curve goes on forever, zero is a non-negative value for y.
Another Example: From the curve y = sin x, you can inspect the range of the function, which is to be between −1 and 1. x can be anything, and in this graph, the domain of y = sin x refers to all values of x. You can keep any number in the sin function, and it will bring an answer between −1 and 1. The range between −1 and 1 will be y, and that means −1 ≤ y ≤ 1.
In this context, we are using only real numbers to define the value of x that can be divided by zero. Imaginary numbers are not applicable. Imaginary numbers are the numbers that we get after applying square root to a negative value.
How to Find Domain and Range Without Using a Graph?
It is easier to find out the domain and range when you are seeing a graph. But it is also important to find out the same without using a graph, and for that, you will have to have a good eye for detail.
If you don’t have graphing software or time to sketch a graph, you must learn how to find out the domain and range without it. So in that case, you check for positive values under the square root sign, and check for the lack of 0 values in the denominator (bottom) of a fraction. Have a look at the example below:
Example:
You will have to find the domain and range of the function without using graphing software or sketching a graph. So, how do we do it? Check out the solution below.
From the example above we can see, there is a square root. We will have to make the values under the square root positive. And for that, we are going to use only x-values that are greater than or the same as -2.
The denominator (bottom) comprises x2−9 that you can write as (x+3)(x−3). So the value of x is not −3 or 3. At the same time, there is no need to worry about the −3 since x≥−2.
Therefore, the domain in this context is x≥−2, x≠3 which can also be [−2,3)∪(3,∞)]. Now find out the range, you will have to consider the upper and bottom of the fraction individually.
Along with the value of x that increased from -2, the top value will increase as well. Now we break the denominator into 4 parts.
When x=-2, the bottom will be (-2)²-9=4-9= -5
Between x=-2 and x=3 (x2−9) becomes a closer value to 0 and f (x) will become −∞ when x=3.
When it is x>3—it’s when x is bigger than 3, so is the value of the bottom, which is more than 0. In this case, f(x) is a great positive number.
If the value of x is significant, the value of the top is large, but the bottom will be more significant. Hence, the function value here is smaller than the top and bottom values. That brings the range to [(−∞,0)∪(∞,0)].
Key Concepts
- The domain of a function is a set of all actual input values that you cannot divide by 0 or a negative number.
- You can graph a piecewise function using the algebraic formula from its assigned subdomains.
- You can use a graph to determine the domain and range of a function in almost all cases.
- You can use the knowledge of toolkit functions to determine the domain and range of relevant functions.
- To identify the domain of a function, identify the input values of the same function that are mentioned as an equation.
Summary
You can usually find the domain by observing the values of the independent variable, which is x in most cases. You can use these values of the independent variable as long as you are avoiding 0 at the bottom of a fraction.
Also, prevent any negative value under the square root symbol. And you can find the range by determining the y-values after substituting them in the potential x-values.