# 1.2 Domain and range

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In this section, you will:
• Find the domain of a function defined by an equation.
• Graph piecewise-defined functions.

If you’re in the mood for a scary movie, you may want to check out one of the five most popular horror movies of all time— I am Legend , Hannibal , The Ring , The Grudge , and The Conjuring . [link] shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain    and range. In this section, we will investigate methods for determining the domain and range of functions such as these.

## Finding the domain of a function defined by an equation

In Functions and Function Notation , we were introduced to the concepts of domain and range . In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.

We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. See [link] .

We can write the domain and range in interval notation    , which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, he or she would need to express the interval that is more than 0 and less than or equal to 100 and write We will discuss interval notation in greater detail later. Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative. #### Questions & Answers How can you tell what type of parent function a graph is ? Mary Reply generally by how the graph looks and understanding what the base parent functions look like and perform on a graph William if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero William y=x will obviously be a straight line with a zero slope William y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis William y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer. Aaron yes, correction on my end, I meant slope of 1 instead of slope of 0 William what is f(x)= Karim Reply I don't understand Joe Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain." Thomas Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-) Thomas GREAT ANSWER THOUGH!!! Darius Thanks. Thomas Â Thomas It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â Thomas Now it shows, go figure? Thomas what is this? unknown Reply i do not understand anything unknown lol...it gets better Darius I've been struggling so much through all of this. my final is in four weeks 😭 Tiffany this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts Darius thank you I have heard of him. I should check him out. Tiffany is there any question in particular? Joe I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously. Tiffany Sure, are you in high school or college? Darius Hi, apologies for the delayed response. I'm in college. Tiffany how to solve polynomial using a calculator Ef Reply So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right? KARMEL Reply The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26 Rima Reply The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer? Rima I done know Joe What kind of answer is that😑? Rima I had just woken up when i got this message Joe Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that Rima i have a question. Abdul how do you find the real and complex roots of a polynomial? Abdul @abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up Nare This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1 Abdul @Nare please let me know if you can solve it. Abdul I have a question juweeriya hello guys I'm new here? will you happy with me mustapha The average annual population increase of a pack of wolves is 25. Brittany Reply how do you find the period of a sine graph Imani Reply Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period Am if not then how would I find it from a graph Imani by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates. Am you could also do it with two consecutive minimum points or x-intercepts Am I will try that thank u Imani Case of Equilateral Hyperbola Jhon Reply ok Zander ok Shella f(x)=4x+2, find f(3) Benetta f(3)=4(3)+2 f(3)=14 lamoussa 14 Vedant pre calc teacher: "Plug in Plug in...smell's good" f(x)=14 Devante 8x=40 Chris Explain why log a x is not defined for a < 0 Baptiste Reply the sum of any two linear polynomial is what Esther Reply divide simplify each answer 3/2÷5/4 Momo Reply divide simplify each answer 25/3÷5/12 Momo how can are find the domain and range of a relations austin Reply the range is twice of the natural number which is the domain Morolake A cell phone company offers two plans for minutes. Plan A:$15 per month and $2 for every 300 texts. Plan B:$25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money? Diddy Reply 6000 Robert more than 6000 Robert For Plan A to reach$27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture