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## Algebra (all content)

### Course: Algebra (all content)>Unit 7

Lesson 5: Introduction to the domain and range of a function

# What is the domain of a function?

Functions assign outputs to inputs. The domain of a function is the set of all possible inputs for the function. For example, the domain of f(x)=x² is all real numbers, and the domain of g(x)=1/x is all real numbers except for x=0. We can also define special functions whose domains are more limited.

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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 ONE number (and ONLY ONE) 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 , 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.
• Is the domain always the x-value
• Yes. If x is the input then the domain is x.
• 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.