How to get the domain and range from the graph of a function
Definition of the domain and range
The domain is all ???x???-values or inputs of a function and the range is all ???y???-values or outputs of a function.
When looking at a graph, the domain is all the values of the graph from left to right. The range is all the values of the graph from down to up.
Basic Functions with Domain Restrictions
Finding the domain and range by looking at the graph of the function
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Domain and range of the graph of the parabola
Example
What is the domain and range of the function? Assume the graph does not extend beyond the graph shown.
Let’s start with the domain. Remember that domain is how far the graph goes from left to right.
Start by looking at the farthest to the left this graph goes. The ???x???-value at the farthest left point is at ???x=-2???. Now continue tracing the graph until you get to the point that is the farthest to the right. The ???x???-value at this point is at ???2???. There are no breaks in the graph going from left to right which means it’s continuous from ???-2??? to ???2???.
Domain: ???[-2,2]??? also written as ???-2\leq x\leq 2???
Next, let’s look at the range. Remember that the range is how far the graph goes from down to up.
Look at the furthest point down on the graph or the bottom of the graph. The ???y???-value at this point is ???y=1???. Now look at how far up the graph goes or the top of the graph. This is when ???x=-2??? or ???x=2???, but now we’re finding the range so we need to look at the ???y???-value of this point which is at ???y=5???. There are no breaks in the graph going from top to bottom which means it’s continuous.
Range: ???[1,5]??? also written as ???1\leq y\leq 5???
Let’s try another example of finding domain and range from a graph.
Example
What is the domain and range of the function? Assume the graph does not extend beyond the graph shown.
Let’s start with the domain. The ???x???-value at the farthest left point is at ???x=-1???. Now continue tracing the graph until you get to the point that is the farthest to the right. The ???x???-value at this point is at ???3???. There are no breaks in the graph going from left to right which means it’s continuous from ???-1??? to ???3???.
Domain: ???[-1,3]??? also written as ???-1\leq x\leq 3???
Next, let’s look at the range. Look at the furthest point down on the graph or the bottom of the graph. The ???y???-value at this point is ???y=0???. Now look at how far up the graph goes or the top of the graph. This is when ???x=3???, but now we’re finding the range so we need to look at the ???y???-value of this point which is at ???y=2???. There are no breaks in the graph going from down to up which means it’s continuous.
Range: ???[0,2]??? also written as ???0\leq y\leq 2???