{"id":1463,"date":"2024-05-24T01:35:26","date_gmt":"2024-05-24T01:35:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1463"},"modified":"2025-08-13T16:13:18","modified_gmt":"2025-08-13T16:13:18","slug":"graphs-and-characteristics-of-basic-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/graphs-and-characteristics-of-basic-functions-learn-it-3\/","title":{"raw":"Domain and Range: Learn It 3","rendered":"Domain and Range: Learn It 3"},"content":{"raw":"

Determine Domain and Range from a Graph<\/h2>\r\nAnother way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the [latex]x[\/latex]-axis. The range is the set of possible output values, which are shown on the [latex]y[\/latex]-axis.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of a polynomial with labels for domain and range[\/caption]\r\n\r\nWe can observe that the graph extends horizontally from [latex]-5[\/latex] to the right without bound, so the domain is [latex]\\left[-5,\\infty \\right)[\/latex]. The vertical extent of the graph is all range values [latex]5[\/latex] and below, so the range is [latex]\\left(\\mathrm{-\\infty },5\\right][\/latex]. Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.\r\n\r\n
Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values.<\/section>
Find the domain and range of the function [latex]f[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of a function[\/caption]\r\n\r\n[reveal-answer q=\"495787\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"495787\"]We can observe that the horizontal extent of the graph is [latex]\u20133[\/latex] to [latex]1[\/latex], so the domain of [latex]f[\/latex]\u00a0is [latex]\\left(-3,1\\right][\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of the previous function with domain and range labeled.[\/caption]\r\n\r\nThe vertical extent of the graph is [latex]0[\/latex] to [latex]\u20134[\/latex], so the range is [latex]\\left[-4,0\\right][\/latex].[\/hidden-answer]<\/section>
Can a function\u2019s domain and range be the same?\r\n<\/strong><\/strong><\/strong>\r\n\r\n
\r\n\r\nYes. For example, the domain and range of the cube root function are both the set of all real numbers.\r\n\r\n<\/section>
Find the domain and range of the function [latex]f[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"489\"]\"Graph Graph of the Alaska Crude Oil production(credit: modification of work by the U.S. Energy Information Administration<\/a>)[\/caption]\r\n\r\n[reveal-answer q=\"834467\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"834467\"]\r\n\r\nThe input quantity along the horizontal axis is \"years,\" which we represent with the variable [latex]t[\/latex] for time. The output quantity is \"thousands of barrels of oil per day,\" which we represent with the variable [latex]b[\/latex] for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as [latex]1973\\le t\\le 2008[\/latex] and the range as approximately [latex]180\\le b\\le 2010[\/latex].\r\n\r\nIn interval notation, the domain is [latex][1973, 2008][\/latex], and the range is about [latex][180, 2010][\/latex]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
[ohm_question hide_question_numbers=1]293805[\/ohm_question]<\/section>","rendered":"

Determine Domain and Range from a Graph<\/h2>\n

Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the [latex]x[\/latex]-axis. The range is the set of possible output values, which are shown on the [latex]y[\/latex]-axis.<\/p>\n

\"Graph
Graph of a polynomial with labels for domain and range<\/figcaption><\/figure>\n

We can observe that the graph extends horizontally from [latex]-5[\/latex] to the right without bound, so the domain is [latex]\\left[-5,\\infty \\right)[\/latex]. The vertical extent of the graph is all range values [latex]5[\/latex] and below, so the range is [latex]\\left(\\mathrm{-\\infty },5\\right][\/latex]. Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.<\/p>\n

Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values.<\/section>\n
Find the domain and range of the function [latex]f[\/latex].<\/p>\n
\"Graph
Graph of a function<\/figcaption><\/figure>\n