How to find domain and range from a graph
A graph gives us two ways to describe where a function lives. We can look left to right to see its inputs, and bottom to top to see its outputs. Let’s read both from the same graph, then try a curve whose lowest point sits at an endpoint.
Look for the x-values and the y-values
The domain is the set of all inputs, or -values, that occur on the graph. Imagine dropping every point on the graph straight down or up onto the -axis. The values that get covered make up the domain. For the range, project the same points horizontally onto the -axis to collect all the outputs.
Find the domain and range of a parabola segment
From left to right, the curve covers every input from to . Both endpoints are included, so the domain is . From bottom to top, the lowest output is at the vertex, and the highest is at both endpoints. Every height between those values occurs, so the range is .
Find the domain and range of an increasing curve
Here the curve starts at and ends at . It covers all the inputs from through , so the domain is . The lowest and highest outputs occur at the two endpoints, and the curve passes through every height in between. That makes the range .
Check which values are actually included
A filled dot includes the point. An open dot excludes that point, so check whether some other included point supplies the same - or -value. An arrow means the graph continues in the direction shown. If the graph skips a stretch of inputs or outputs, list the separate intervals. For example, includes both endpoints of the first interval, skips , and includes .
