Problem 115 - College Algebra Essentials

Question

Explain how to identify the domain and range of a function from its graph.

Step-by-Step Solution

Answer

The domain and range of a function from its graph can be identified by analyzing the horizontal and vertical extent of the graph, respectively. The domain is determined by the x-coordinates of the leftmost and rightmost endpoints of the graph, and the range is determined by the y-coordinates of the lowest and highest points of the graph.

Step 1: Understanding Domain and Range

The 'domain' of a function is the set of all possible input values (often denoted as 'x') which the function can handle. This corresponds to the horizontal extent of the graph. The 'range', on the other hand, is the set of all possible output values (often referred to as 'y') that function can produce. This corresponds to the vertical extent of the graph.

Step 2: Identifying the Domain from the Graph

In a graph, to find the domain (possible 'x' values), look for the leftmost point and the rightmost point where the function exists. The x-coordinates of these points give you the endpoint of the domain. For most typical functions, the graph extends infinitely to the left and right, so the domain is all real numbers. However, if there are vertical lines (asymptotes) that the graph doesn't cross, or if the graph doesn't extend forever, the domain will be a specific interval or set of intervals.

Step 3: Identifying the Range from the Graph

Finding the range (possible 'y' values) from a graph requires looking for the lowest and highest points of the graph. The y-coordinates of these points give the boundaries of the range. Again, similar to the domain, many functions might have the range of all real numbers. If there are horizontal asymptotes that the graph doesn't cross, or if the function doesn't extend forever up or down, the range will be a specific interval or set of intervals.