Domain and Range of Graphs of Functions

Section 2.5: Domain and Range of graphs of functions Chapter 2: Functions, Linear equations, and inequalities Determine if a graph represents y as a function of x. We learned how to determine if a relation represents y as a function of x in the previous section. 1) 𝑦 is a NOT a function of 𝑥 if there are two or more points with the same 𝑥 − 𝑣𝑎𝑙𝑢𝑒, but 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠. 2) 𝑦 is a function of 𝑥 if each 𝑥 has a different 𝑦. To determine if a graph represents 𝑦 as a function of 𝑥 can be difficult using what we know so far. This is because points on a graph are not always marked. This can make it hard to find points that have the same 𝑥 − 𝑣𝑎𝑙𝑢𝑒𝑠 with different 𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠 . There is technique called the vertical line test that is often used to determine if a graph represents y as a function of x. The vertical line test is a quick way to determine if a graph represents 𝑦 as a function of 𝑥 without the need of listing different points with the same 𝑥 − 𝑣𝑎𝑙𝑢𝑒𝑠 but different 𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠. Vertical line test: • If a vertical line can be drawn to touch the graph of a function in more than one place, then 𝑦 is NOT a function of 𝑥. • If it is not possible to draw a vertical line to touch the graph of a function in more than one place, then y is a function of x. For Example: Use the vertical line test to determine if the graph depicts y is a function of x. Solution: y is NOT a function of x (as it is possible to draw a vertical line to touch the graph in more than one place.) Notice, the vertical line test has found two points on the graph { (4,2) and (4,-2)} that have the same x with different y’s. Section 2.5: Domain and Range of graphs of functions Chapter 2: Functions, Linear equations, and inequalities For Example: Use the vertical line test to determine if the graph depicts y is a function of x. Solution: y is a function of x (as no vertical line can be drawn to touch the graph in more than one place.) Section 2.5: Domain and Range of graphs of functions Chapter 2: Functions, Linear equations, and inequalities Find the Domain and Range from the graph of a continuous function using INTERVAL NOTATION Interval notation has two types of symbols: 1) Parenthesis ( ) Use round parenthesis when: a) point is marked with an open circle ○ b) value is infinity ∞ 2) Bracket [ ] Use bracket when: a) point is marked with a closed circle b) point is an unmarked point on a graph Writing the domain and or range may require a bracket on one side of the interval and a parenthesis on the other. • How to find domain from a graph of a continuous function and write answer in interval notation. Step 1) Identify the x-coordinate of left-most the point on the graph. Place it after the appropriate symbol ( or [. Step 2) Identify the x-coordinate of the right-most point on the graph. Place it before the appropriate symbol ) or ]. • How to find range from a graph of a continuous function and write answer in interval notation. Step 1) Identify the y-coordinate of lowest point on the graph. Place it after the appropriate symbol ( or [. Step 2) Identify the y-coordinate of the highest point on the graph. Place it before the appropriate symbol ) or ]. It is common for a to graph continue beyond the portion of the graph we can see; the domain and range may be greater than the visible portion of the graph. We generally need ∞ for one or both sides of the interval of the domain and or range when a graph extends beyond what we can see. Section 2.5: Domain and Range of graphs of functions Chapter 2: Functions, Linear equations, and inequalities Find the domain and range of the graph below, write answer in interval notation. Range: First: Find the y-coordinate of the bottom point and decide whether to put a ( or [ before the number. The bottom point is (-3,-5). The ycoordinate of the top point is y = -5 Domain: First: Find the x-coordinate of the point that is furthest left and decide whether to put a ( or [ before the number. The point that is furthest left is (-3, -5). The xcoordinate of the point is x = -3. “[“ is needed as the point is marked with a closed circle. The domain will start with [-3, Second: Find the x-coordinate of the point that is furthest right and decide whether to put a ) or ] after it. The point that is furthest right is (2,0). The xcoordinate is x = 2. “[“ is needed as the point is marked with a closed circle. The range will start with [-5 Second: Find the y-coordinate of the top point and decide whether to put a ) or ] after it. The top point is (0,4). The y-coordinate of the top point is y = 4. “ ]” will be needed to end the range as the point (0,4) is an unmarked point on the graph. y to end the range: ,4] Answer: Range [-5, 4] “)” is needed as the point is marked with an open circle. The domain will end with ,2) Answer: Domain [-3,2) Section 2.5: Domain and Range of graphs of functions Chapter 2: Functions, Linear equations, and inequalities For Example: Find the domain and range of the graph below, write answer in interval notation. Range: First: Find the y-coordinate of the bottom point and decide whether to put a ( or [ before the number. This is a situation where the graph continues beyond what can be seen. Domain: First: Find the x-coordinate of the point that is furthest left and decide whether to put a ( or [ before the number. The point that is furthest left is (2, -1). The xcoordinate of the point is x = 2. ” [“ is needed as the point is marked with a closed circle. The graph extends to the bottom of the y-axis. When a graph extends to the bottom of the yaxis, −∞ will be needed to start the range. “(“is needed: round parenthesis are always used for ∞ 𝑎𝑛𝑑 − ∞. The range will star with (−∞, Second: Find the y-coordinate of the top point and decide whether to put a ) or ] after it. The domain will start with [2, Second: Find the x-coordinate of the point that is furthest right and decide whether to put a ) or ] after it. This is a situation where the graph continues beyond what can be seen. The graph extends to the far-right edge of the xaxis. When a graph extends to the far-right edge of the x-axis, ∞ will be needed to end the domain. ” )” is needed: round parenthesis are always used for ∞. The domain will end with , ∞) Answer: Domain [-1,∞) The top point is (2,-1). The y-coordinate of the top point is y = -1. ” ]” will be needed to end the range as the point (2,-1) is marked with a closed circle. y to end the range: ,-1] Answer: Range (−∞, −1] Section 2.5: Domain and Range of graphs of functions Chapter 2: Functions, Linear equations, and inequalities Find the domain and range of the graph below, write answer in interval notation. Range: First: Find the y-coordinate of the bottom point and decide whether to put a ( or [ before the number. This is a situation where the graph continues beyond what can be seen. Domain: First: Find the x-coordinate of the point that is furthest left and decide whether to put a ) or ] after it. This is a situation where the graph continues beyond what can be seen. The graph extends to the far-left edge of the x-axis. When a graph extends to the far-leftt edge of the xaxis, −∞ will be needed to start the domain. ” (” is needed: round parenthesis are always used for ∞. The domain will start with (-∞ Second: Find the x-coordinate of the point that is furthest right and decide whether to put a ) or ] after it. This is a situation where the graph continues beyond what can be seen. The graph extends to the bottom edge of the y-axis on both sides of the graph. When a graph extends to the bottom of the y-axis, −∞ will be needed to start the range. “(“is needed: round parenthesis are always used for ∞ 𝑎𝑛𝑑 − ∞. The range will start with (−∞, Second: Find the 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the top point and decide whether to put a ) or ] after it. The top point is (0,2). The 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the top point is 𝑦 = 2. ” ]” will be needed to end the range as the point (0,2) is unmarked. y to end the range: , 2] Answer: Range (−∞, 2] The graph extends to the far-right edge of the x-axis. When a graph extends to the far-right edge of the xaxis, ∞ will be needed to end the domain. ” )” is needed: round parenthesis are always used for ∞. The domain will end with , ∞) Answer: Domain (−∞, ∞) Section 2.5: Domain and Range of graphs of functions Chapter 2: Functions, Linear equations, and inequalities Find the domain and range of the graph below, Range: write answer in interval notation. First: Find the 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the bottom point and decide whether to put a ( or [ before the number. The bottom point is (3,-2). The 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the bottom point is 𝑦 = −2 “(“ is needed as the point is marked with an open circle. Domain: First: Find the 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the point that is furthest left and decide whether to put a ( or [ before the number. The point that is furthest left is (−3, 4). The xcoordinate of the point is 𝑥 = −3. “(“ is needed as the point is marked with an open circle. The domain will start with (−3, Second: Find the 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the point that is furthest right and decide whether to put a ) or ] after it. The point that is furthest right is (3, −2). 𝑇ℎ𝑒 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 is 𝑥 = 3 The domain will end with , 3) “)” is needed as the point is marked with an open circle. Answer: Domain (−3, 3) The range will start with (−2 Second: Find the 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the top point and decide whether to put a ) or ] after it. The top point is (−3,4). The 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the top point is 𝑦 = 4. “ )” will be needed to end the range as the point (0,4) is marked with an open circle. 𝑦 to end the range: , 4) Answer: Range (-2, 4) Section 2.5: Domain and Range of graphs of functions Chapter 2: Functions, Linear equations, and inequalities Find the domain and range of the graph below, write answer in interval notation. Range: First: Find the y-coordinate of the bottom point and decide whether to put a ( or [ before the number. This is a situation where the graph continues beyond what can be seen. Domain: First: Find the x-coordinate of the point that is furthest left and decide whether to put a ) or ] after it. This is a situation where the graph continues beyond what can be seen. The graph extends to the far-left edge of the x-axis. When a graph extends to the far-left edge of the xaxis, −∞ will be needed to start the domain. ” (” is needed: round parenthesis are always used for ∞. The domain will start with (-∞ Second: Find the x-coordinate of the point that is furthest right and decide whether to put a ) or ] after it. This is a situation where the graph continues beyond what can be seen. The graph extends to the far-right edge of the x-axis. When a graph extends to the far-right edge of the xaxis, ∞ will be needed to end the domain. ” )” is needed: round parenthesis are always used for ∞. The domain will end with , ∞) Answer: Domain (−∞, ∞) The graph extends to the bottom edge of the y-axis on both sides of the graph. When a graph extends to the bottom of the y-axis, −∞ will be needed to start the range. “(“is needed: round parenthesis are always used for ∞ 𝑎𝑛𝑑 − ∞. The range will start with (−∞, Second: Find the y-coordinate of the top point and decide whether to put a ( or [ before the number. This is a situation where the graph continues beyond what can be seen. The graph extends to the top edge of the yaxis on both sides of the graph. When a graph extends to the bottom of the y-axis, ∞ will be needed to end the range. “(“is needed: round parenthesis are always used for ∞ 𝑎𝑛𝑑 − ∞. The range will end with ,∞) Answer: Range (−∞, ∞) Section 2.5: Domain and Range of graphs of functions Chapter 2: Functions, Linear equations, and inequalities #1-10: Use the vertical line test to determine if the graph represents y as a function of x. 1) 2) 3) 4) 5) 6) Section 2.5: Domain and Range of graphs of functions Chapter 2: Functions, Linear equations, and inequalities #1-10 Continued: Use the vertical line test to determine if the graph represents y as a function of x. 7) 8) 9) 10) Section 2.5: Domain and Range of graphs of functions Chapter 2: Functions, Linear equations, and inequalities #11 – 30: Find the domain and range of function. Write your answer in interval notation. 11) 12) 13) 14) 15) 16) Section 2.5: Domain and Range of graphs of functions Chapter 2: Functions, Linear equations, and inequalities #11 – 30 Continued: Find the domain and range of function. Write your answer in interval notation. 17) 19) 18) 20) Section 2.5: Domain and Range of graphs of functions Chapter 2: Functions, Linear equations, and inequalities #11 – 30 Continued: Find the domain and range of function. Write your answer in interval notation. 21) 23) 22) 24) Section 2.5: Domain and Range of graphs of functions Chapter 2: Functions, Linear equations, and inequalities #11 – 30 Continued: Find the domain and range of function. Write your answer in interval notation. 25) 26) 27) 29) 28) 30)