Domain and range of a function help

[Rhine720]

So I'm skipping pre-cal with full intentions of picking up on AP Calc. Our summer course work is taken from a book called preparing for calculus. After giving me some who-dad that I knew already everything changed in an instant. There's a little gap that isn't quiet filled.

A function is still a linear equation. So what's the difference between plotting data predictions int he form of a line equation and plotting anything as a function?

Also when we find the domain and range of a function. Don't we do things like that when we set up T-tables,assign a random value to x and use it to solve for y? I know the domain and range account for all the the numbers that will work, but isn't the x we get from a t-table int he domain of a function(providing it gives an answer once substituted in an equation).

 

[jegues]

A function is still a linear equation.

A function is only a linear equation if it is defined in that manner.

So what's the difference between plotting data predictions int he form of a line equation and plotting anything as a function?

I don't understand your reasoning behind this question, so I'll just throw out some basic intution when thinking about functions.

Think of a function as a magical black box, a box so magical that when you put a number in this box, let's say for example 2, the box spits out a 3.

So you scratch your head and say well what I put a 6 in the box? So you put 6 inside the magical black box and it spits out a 7.

Then you say aha! I'll trick this magical box by putting in a variable, x, instead of number and see what I get.

So you do and you end up getting x + 1. (e.g. y = x + 1)

Hopefully that gives you some sort of intution behind functions.

domain and range of a function.

The domain is simply the set of values of the independent variable (x) for which a function is defined.

The range is simply the set of values of the dependent variable (y) for which a function is defined.

Hopefully this is some use to you.

 

[Mark44]

So I'm skipping pre-cal with full intentions of picking up on AP Calc. Our summer course work is taken from a book called preparing for calculus. After giving me some who-dad that I knew already everything changed in an instant. There's a little gap that isn't quiet filled.

A function is still a linear equation.

No. A function is a pairing of elements in one set with elements in another set so that each element in the first set (the domain) gets paired with one element in the second set (the range or co-domain). A function might be defined by an equation, but there is no requirement that the equation be linear. For example, consider f(x) = x². This is not a linear equation.

So what's the difference between plotting data predictions int he form of a line equation and plotting anything as a function?

As already noted, the equation for a function does not have to be a linear equation, so its graph will not be a straight line.

Also when we find the domain and range of a function. Don't we do things like that when we set up T-tables,assign a random value to x and use it to solve for y? I know the domain and range account for all the the numbers that will work, but isn't the x we get from a t-table int he domain of a function(providing it gives an answer once substituted in an equation).

When you make a table of x- and y-values for a function, you normally pick values of x that are in the domain, but you probably don't do any further analysis to say exactly which values of x make up the domain, and you probably don't try to say what values are in the range.

What you're being exposed to goes much further than determining a few pairs of values in a table.

For example, suppose that
$$f(x) = \frac{1}{\sqrt{x^2 - 1}}$$

You probably won't be able to figure out the domain by just picking values of x at random, but in this class you will work with this equation to find that the domain is in two disjoint pieces and the the range is all positive real numbers.

 

[Rhine720]

I'm sorry I didn't mean linear. Heh, I had a feeling I was wording everything all wrong.

so is F(x)=x² the same as y=x²?
or F(x)=mx+b is the same as y=mx+b ? Also any other graph-able equation you can think of.

" A real-valued function f of a real variable x from X to Y is a correspondence that assigns to each number x in X exactly on number y in Y. "

So in a t-table when x = a value, it will also give the value of y. And unless the equation changes this will be the only number that makes y equal that ( i know there are other cases)

Am I getting this right? I fully understand finding the the domain and range. But I'm tyriing to understand this little concepts. I

 

[Mark44]

I'm sorry I didn't mean linear. Heh, I had a feeling I was wording everything all wrong.

so is F(x)=x² the same as y=x²?
or F(x)=mx+b is the same as y=mx+b ? Also any other graph-able equation you can think of.

The equations with F(x) use function notation and have the same graphs as their counterparts that have y in terms of x.

So in a t-table when x = a value, it will also give the value of y. And unless the equation changes this will be the only number that makes y equal that ( i know there are other cases)

It depends on the function. For example, if f(x) = x², f(2) = 4 and f(-2) = 4. There are two x values that are paired with a y value of 4.

Am I getting this right? I fully understand finding the the domain and range. But I'm tyriing to understand this little concepts. I

 

[QuarkCharmer]

I know the domain and range account for all the the numbers that will work

Well, I believe the domain is the only set that accounts for all numbers that will work. The range does as well I guess, but it is totally dependant on the domain. Domain=Input Range=Output

Unless you look at it backwards, and switch X and Y. In that case the function would work the opposite way, and the range becomes the domain, so it's still the same thing.

When I was doing this, we were using the zero-product property, and simplifying expressions (symbolically), that contained functions themselves.

Functions of a Function
Ex: f(x) = x² + 1 and g(x) = 5x.
f(g(x)) = f(5x) = (5x)² + 1 = 25x² + 1

Zero-Product Property to determine the domain
For instance, f(x)=(2+x)/((5-x)(x+5))
If you set the denominator to 0,
(5-x)(x-5)=0
You know that one of those must be equal to 0.
5-x=0 x-5=0
x=-5 x=5
Now since you cannot divide by 0, you know the denominator cannot be 5 or -5.
Therefor, the domain of the function is all real numbers except 5 and -5.
In interval notation
(-∞,-5)U(-5,5)U(5,∞)

Source: http://www.themathpage.com/aprecalc/functions.htm
(That link is pretty neat, in that, you can work along with it and the answers are hidden until you mouse-over)