One way to determine the domain and range of a certain function is by using graphs. The domain refers to the set of all possible x-values, while the range refers to the set of all possible y-values.
For example:
To determine the domain of this function, we just have to take a look at all the possible x-values.
Since this graph continues to infinity without any restrictions, we can say that the domain for this function is all real numbers. We can write it as:
[tex]\begin{gathered} \text{ Domain:}(-\infty,\infty) \\ \text{ Domain: }\lbrace x|x\in R\rbrace \end{gathered}[/tex]
For the range, we will now look at all possible y-values. As we can see from the graph, the graph only starts at (-2, 1) and above. This would mean that the only possible values of y would be from 1 up to positive infinity.
We could write this as:
[tex]\begin{gathered} \text{ Range: }[1,\infty) \\ \text{ Range: }\lbrace y|y\ge1\rbrace \end{gathered}[/tex]
Now, what if a graph is not provided and only an equation is given. The fastest way to solve for the domain and range of a given equation is to solve for any restrictions that are present in the function.
For example:
[tex]y=2x+3[/tex]
Here, we can see that we can input any x-values and it will give us a y-value.
Therefore, we can easily say that the domain and range for this function are:
[tex]\begin{gathered} \text{ Domain:}\lbrace x|x\in R\rbrace \\ \text{ Range: }\lbrace y|y\in R\rbrace \end{gathered}[/tex]
How about this function:
[tex]y=\frac{2}{x+3}[/tex]
For this example, the restriction that we might think of is that the denominator of the rational expression cannot be equal to 0. Therefore, we need to find that x-value that will make our rational equal to 0.
For that, we could just take the denominator, equate it to 0, and then solve for x.
[tex]\begin{gathered} x+3=0 \\ x=-3 \end{gathered}[/tex]
Therefore, any x-value would be possible except for x = -3. We can now write the domain and range as:
[tex]\begin{gathered} \text{ Domain: }\lbrace x|x\ne-3\rbrace \\ \text{ Range: }\lbrace y|y\in R\rbrace \end{gathered}[/tex]
We knew that the range of this function would be any real numbers since any input in the y-value would give us a real x-value without any restrictions. Therefore, the function would work for any y-value as long as it is a real number.
