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Understanding Domains and Ranges, Mathematical Functions

Review how to find domain and range of functions, including numeric ordering and restrictions. Learn to identify domains for various types of functions and relations. Practice determining domains based on real-world scenarios.

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Understanding Domains and Ranges, Mathematical Functions

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  1. Functions:Domain and Range

  2. Review: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 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}

  3. 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) = • 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) = • 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

  4. Find the domain of the relation: Domain: {-2 < x < 2} Domain: {all real numbers} • 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

  5. Find the 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: 6 am, 82°F; 8 am 78°F; 9 am 76°F; 10 am, 74°F, and 11 am, 72°F. • Domain: {6, 8, 9, 10, 11} • Range:{72, 74, 76, 78, 82}

  6. 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!

  7. 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

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