Math CentralQuandaries & Queries


Question from Dawn, a student:

Hi, I've been out of school for 8 years and recently picked up a math correspondence course. I'm having trouble trying to figure out the range and domain of a linear function. I've read everything I can find in my text, lessons and on your site and I still can't figure out what I'm suppose to be doing. The function is y=2x+1. Please help.


At times it helps to think of a numeric function as a process of machine. You input a number to the machine, some processing goes on and the machine outputs a number. From this point of view the domain is all the numbers you can input to the machine and receive a valid output and the range is all the possible outputs from the machine. So, for example consider y = √x. There is a button your calculator that is precisely this function machine. you enter a number and then push the √ button to activate the process. If you input a positive number or zero the process returns the square root of the number. If you input a negative number you get an error message. The domain of the √ function is all non-negative numbers.

Another example that might be a button your calculator is y = x2. This time you can input any number and calculate its square so the domain is all numbers. The output in this case is never negative since a square can't be negative. Hence the range of this function is all non-negative numbers.

Your function is y = 2x + 1. Is there an input x that causes this function to produce an error message? No! For any number x you double it and then add 1 which you can always do so the domain of the function is all numbers. What about the range? The range is usually harder but in this case it is straightforward. Can you get every number y as an output of this function? Yes you can. If you give me a value for y, say y = b then I look at b = 2x + 1 and solve for x. This gives x = (b - 1)/2 so this value of x when input into the function y = 2x + 1 gives b as an output. Thus the range of y = 2x + 1 is also all numbers.

I hope this helps,

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