Domain and Range

Vocabulary Independent Dependent Domain Range Domain: All of the input values, x-values, independent values. The independent quantity is the quantity that stands alone and is not changed by the other quantities. To State Domain: list all x values separated by commas in brackets { } Domain {-5,0,5} Range: All of the output values, y-values, dependent values. The dependent quantity depends on the independent quantity. The independent quantity causes a change in the dependent quantity To State Range: list all y values separated by commas in brackets { } Range {10,0,10}

State the domain and range of the following relations: (x, y) Set W = {(3, 4), (5, 9), (-6, 4), (8, 12) Domain: {-6, 3, 5, 8} Range: {4, 9, 12} Is this a function? (grade, student name) Set A = {(A, Frankie), (C, Denise), (B, Brady), (A, Julietta), (D, Bob), (B, Evelyn)} Domain: {A, C, B, D} Range: {Frankie, Denise, Brady, Julietta, Bob, Evelyn} *tip – you should put the entries in numeric order and not list duplicates but it means the same thing as domain {3,5,-6, 8} range {4, 9, 4, 12} YES! Function NO! Not a function b/c A input had two different outputs

Practice: Find the domain and range of the function: Domain: {list all x values in table} in numeric order without listing repeats Range: {list all y values in table} in numeric order without listing repeats x f(x) 10 -9 8 3 4 -5 -12 -15 x f(x) 3 5 -2 6 -4 10 -12 12 -15 This is the same as Domain: {0, -9, 3, 4, -12} Range: {10, 8, 10, -5, -15} you might see it written this way sometimes but numeric order is better Domain: {-12, -9, 0, 3, 4} Range: {-15, -5, 8,10} Domain: {3, 5, 6, 10, 12} Range: {-15, -12, -4, -2, 5}

State the domain and range of the following graphs: Discrete graphs: list the x and y values. Continuous graphs: describe the intervals or ranges of x and y values (inequalities) *check for arrows on the end! Domain: {All real numbers} Range: {y ≥ -4} Is it a function? Domain: {-1, 1, 3, 4, 6} Range: {-3, 1, 3, 4} Is it a function? Domain: {2} Range: {All real numbers} Is it a function? YES! Function YES! Function NO! Not a function *tip – it’s normal to put the entries in numeric order but it’s the same as range {1, 4, -3, 3}

Find the domain of the relation: Domain: {all real numbers} Domain: {-2 < x < 2} Even though you can’t see all x values of this function from the graph, it will go on forever in each direction (we will learn more about identifying that later) This is NOT a function, but the relation still has a domain and range Domain: {x > 0} Domain: {all integers} Notice this ends up being a discrete graph

Identify the dependent and independent quantities Your grade on a test and the number of hours you studied. The dependent quantity is your grade (because your grade depends on how long you studied). This is the range, output, y values The independent quantity is the number of hours you studied. This is the domain, input, x values Caroline makes $8.50 an hour babysitting for her neighbors’ children after school and on the weekends. The dependent quantity is the total amount of money Caroline earns (because her pay depends on how long she works) The independent quantity is the total number of hours she babysits

Identify the dependent and independent quantities Pedro is hiking in a canyon. At the start of his hike, he was at 3500 feet. During the first 20 minutes of the hike, he descended 500 feet at a constant rate. Then he rested for half an hour before continuing the hike at the same rate. The dependent quantity is the elevation (because his elevation depends on how long he has hiked) This is the range / output / y values and will be graphed on the y axis The independent quantity is time (how long he has been hiking) This is the domain / input / x values and will be graphed on the x axis

Find the dependent and independent quantities and domain and range of the function: The temperature in a house drops 2°F for every hour the air conditioner is on between the hours 6:00 am and 11:00 am. The following is a list of times and temperatures in the house: 6am, 82°F; 8am 78°F; 9am 76°F; 10am, 74°F; 11am, 72°F Dependent: temperature Independent: time Domain: {6, 8, 9, 10, 11} Range: {72, 74, 76, 78, 82}

Find the domain of the function: If the function f(n) represents the number of hours required to construct n pizzas at dinner time at the local delivery joint, what domain makes sense? You can’t make negative pizzas or partial pizzas. Domain: {whole numbers ≥ 0} Will the graph of this be discrete or continuous? Discrete!

Find the domain of the function: Your cell phone plan charges you $0.20 for each text message you send. Your parents put a cap of $50 on your texting bill each month. If f(x) = 0.2x is the cost of the total number of texts you send per month, what is the domain of the function? First find the maximum amount of texts you can send each month by substituting 50 in where f(x) is. 50 = 0.2x x = 250 You cannot go over 250 texts. Can you have negative text messages? Write the domain. Domain: {0 ≤ x ≤ 250} So the largest amount you can have is 250 No, so the smallest amount you can have is 0

Find the domain of the function: f(x) = 3x + 9 If there aren’t exact values to list, describe the domain The domain (x values) could be any real number. D = {all real numbers} f(x) = 𝒙 −𝟓 You can’t take the square root of a negative number, but 0 and larger is ok The domain is any real number ≥ 5. D = {All real numbers ≥ 5 } or could write it as D = {x > 5} f(x) = 𝟑𝒙 𝒙−𝟒 The denominator cannot equal zero. The domain is any real number except 4. D = {All real numbers ≠ 4} *Tip – to find the number it can’t be or what it should be greater or less than, set the expression inside = 0 or denominator = 0 and solve x - 4 = 0 x = 4 , so x can’t be 4