How to find domain of a function

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  I really didn't get the x€R thing. It didn't make any sense. We already covered this in school and it never came up. What is it?

  • The ϵ is the Greek letter epsilon. In this context epsilon is a symbolic way of saying "member of", so when you see "x ϵ" it means "x is a member of."

    Suppose you were a part of a cupcake club called "The Yummies". Then 4Abbyrose1 ϵ The Yummies would read 4Abbyrose1 is a member of The Yummies. Now while you wouldn't use ϵ when you are writing English, you use it a lot when writing math - and there are many many more symbols that are a kind of "math Shorthand".

    Often you will see something like x ϵ R, which means "x is a member of the real numbers", or you can just say "x is a real number". If you saw n ϵ Z, that means n is a member of the integers, or you can just say n is an integer.

    Remember when you used "x" as the multiplication sign? - Well, ϵ may seem weird now, but later, as you become more used to it, you will appreciate it. If you continue in math, you'll meet it often.

• Sal says that g(y) = (y-6)^1/2.
  Let y be 15.
  (15-6)^1/2
  (9)^1/2
  or +3 or -3. But as per definition can a function have 2 values for a given input?

  • 
    

    Good question.
    When dealing with a square root function, we only consider the principal root (the positive root).

• If division by zero is undefined,then wouldn't the square root of zero also be undefined??In your example you give yis greater than or "equal" to six.If y=6 then you end up with the squared roo of zero,don't you??

  • 
    

    Division by zero IS NOT defined because there's NO number that when multiplied by zero gives the original (non-zero) number that's being divided by zero.
    But there is ONLY ONE number that when multiplied by itself gives the answer of zero --- and that is zero. So the square root of zero IS defined and is zero.

• I'm still confused, how do you know if the domain or range is a real number?

  • 

    Any number in the coordinate plane is a real number.

• For the second example to the video, at approximately

  4:19

  , I'm wondering why negatives aren't possible with the square root sign. I can see how it has to be greater than zero, its only the greater than six that confuses me.

  • the value of y has to be greater than or equal to 6, or else the number under the radical would become negative. As we know, the square root of a negative number is not real, thus y cannot be less than 6.

    For eg, for y = 5, g(y) = (-1)^1/2, which is not a real number. Therefore, y >= 6.

• What are the definitions of real numbers?

  • The type of number we normally use, such as 1, 15.82, −0.1, 3/4, etc.
    Positive or negative, large or small, whole numbers or decimal numbers are all Real Numbers.
    They are called "Real Numbers" because they are not Imaginary Numbers.

• This concept makes sense to me when Sal does it with smaller numbers but I'm having trouble with a specific homework problem and I'm not sure how to use what Sal teaches in this video to go about solving it...

  The question is as follows:

  g(x)= x+1/x^2-2x-15

  (function of x equals 1 plus x all over x squared minus 2x minus 15)

  Any help or tips would be super appreciated!!

  • 

    g(x) = (x + 1) / (x^2 - 2x - 15)

    The function will be defined for all real numbers except when the denominator equals 0. (We cannot divide by 0 after all.) The easiest way to determine when the denominator equals 0 is to factor the quadratic equation.

    g(x) = (x + 1) / ((x - 5) * (x + 3))

    As you can see, the denominator will be 0 when x = -3 or x = 5. So, the domain of g is all real numbers except -3 and 5.

• Is the domain always the x-value

  • Yes. If x is the input then the domain is x.

• G (y) =√ (y-6) has a sqr so it going to have 2 outputs, does it still being a function??

  • When dealing with square root functions, we only deal with principal roots (meaning just the positive value).

• Why did Sal describe the function g(y) in terms of real numbers? I mean that we do know about complex numbers and the imaginary number i, which is useful for finding square roots of negative numbers. Why didn't he simply consider y belonging to the set of Complex numbers? Was it to make the function look easier and to make it understandable by other people?

What is the easiest way to find the domain?

Which is the domain in a function?