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Explain what domain and range are,

Under what circumstances will a function have domain other
than all real numbers?

Provide an example of a function whose domain is all real
numbers and explain why. Your example can be either a graph or an equation.

Provide an example of a function whose domain isn't all real
numbers and explain why. Your example can be either a graph
or an equation.

Provide a third example to find the domain of.

 

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2 Answers

Hi Crystal!
 
So basically, the domain is the set of all possible inputs for a function, while the range is the set of all the possible output for the function (This is a VERY basic definition. When you get into higher maths, you'll discover that domain and range are two sets that have some kind of mapping between them.). There are many instances when the domain isn't all the reals. Bu first lets talk a little about functions.
 
I doubt you've seen it presented like this (this is one of the many, many pitfalls of the American education system) is to think of a function as a machine. Think about a toaster, for example. When we put something in the toaster, the toaster transforms it into something else (this is our function rle).We can put bread or pop-tarts in a toaster, and we will get out toast or an even more yummy pop-tart. But no matter how much we would like to, we can't put an egg in our toaster. If we do that, the machine breaks and our house blows up. So for our toaster, we say that the domain (the things that we can put in) includes bread and a pop-tart. We say that our range (the things we get out) includes the toast and the even more yummy pop-tart. So, our domain isn't any food we like. So lets consider a basic function:
 
y=5x+3
 
No matter what number we put into this equation, our house won't blow up. But let's look at another function:
 
y=1/x
 
Can you think of any values you can put in that would make this function break?
 
X CANNOT BE 0! Yeah we can't divide by zero. So for this function, our domain is NOT all real numbers. Anytime you have a function that has an expression involving x in the denominator, set the denominator equal to 0 and solve for x. This value of x cannot possibly be in our domain (There are sometimes when this is not true. When this is not true, we say that the function has a hole in it's graph.). Now let's consider another function:
 
y=sqrt(x+2)
 
What kind of numbers can we not take the square root of?
 
We cannot take the square root of negative numbers. So we need x+2 to be positive:
 
x+2>=0
x>=-2
 
So the domain of our function is all real numbers greater than or equal to -2.
 
The problem I am going to give you is significantly harder than the problems I presented above, but it relies on the exact same concepts. So try your best, and present your solution on this forum chain.
 
Find the domain of f(x)=1/[sqrt(x^2 -9)].
 
Basically, for a function, you can think of domain as "what you can put in that will result in an answer" or the x's and range is "what you get out", the y's.  Another way to define this would be that the domain is all the possible inputs, and the range is all the possible outputs.
 
For an example, lets use the function y=x2.
 
The domain of this function is all the real numbers.
Why? Because you can have any real number be x and you will get an output.
Some examples:
If x=0, then y=(0)2=0
If x=4, then y=(4)2=16
If x=-9, then y=(-9)2=91
If x=4.34, then y=(4.34)2=18.8356
and so on.
And it can be seen that the range of this function is all positive real numbers.
 
Now, lets change the function just a bit, y=1/x2
 
The domain is now all real numbers except for 0.
Why? Because when x=0, the function is not defined, therefore, there is no result.
 
Another example of this would be a function similar to y=1/(x-3)
 
In this case, the domain is all real numbers except 3.
Why? Again, if x=3, the denominator would be 0, and you would not get a valid result.
 
Hope this helps!