Functions Interpretation of

[Probability]

I have read my course book quite a few times now to try and get an understanding of functions and intervals and must say that I don't get it?

I understand equalities and the difference between open and closed, or half open and half closed intervals, but the part where I get lost is;

example;

g(x) = Sqrt x

The function g has domain {0, infinity} since square root is only defined for x greater than or equal to 0.

So I have these intervals, closed, open and half open and half closed, the example above says g = {0, infinity}

How would I know how a domain of a function is equal to any one type of interval?

I could be asked say;

h(x) = 1/x

How would I know what the domians would be and how do I select the right type of intervals?

I am lost with this?

Any help appreciated.

 

[pka]

I understand equalities and the difference between open and closed, or half open and half closed intervals, but the part where I get lost is;
example;
g(x) = Sqrt x
The function g has domain {0, infinity} since square root is only defined for x greater than or equal to 0.

I could be asked say;
h(x) = 1/x

I agree with Jeff. Your notation is anything but standard.

The domain of \(\displaystyle \sqrt{x}\) is \(\displaystyle [0,\infty)\) which is a closed interval.

The domain of \(\displaystyle \frac{1}{x}\) is \(\displaystyle (-\infty,0)\cup(0,\infty)\) which is the union of two open intervals.

 

[Probability]

I am not familiar with the notation that you are using. And I am not sure that I understand the full extent of your question, but let's try.

A function is just a rule or mapping UNAMBIGUOUSLY taking an input number (or ordered set of numbers) to an output number (or ordered set of numbers). Nothing complicated about it. It is usually implicitly understood that the numbers being discussed are real numbers. OK so far?

In the absence of any special information further restricting the domain, the domain is assumed by default to be all real numbers for which the function gives an output number that is a real number.

Let's use your example of \(\displaystyle \sqrt{x}\).

If x is negative, then \(\displaystyle \sqrt{x}\) is not a real number. So negative real numbers are excluded from the domain. You understood this.

Is zero excluded from the domain? No \(\displaystyle \sqrt{0} = 0\ because\ 0 * 0 = 0\).

So the domain is the set of all non-negative real numbers. Make sense so far?

Now the notation that I learned to show that is \(\displaystyle [0, \infty)\). The interval is semi-open because it includes one real endpoint, namely 0, but has no real endpoint on the other side.

OK Now let's have you do a bit of work.

Are there any real numbers x such that \(\displaystyle \dfrac{1}{x}\) is not a real number? What are they?

Jeff thanks for your thread,this subject is brand new to me although I have seen it before have never read into the understanding of it. My coursebook appears to skim the surface of this subject and puts people like me in a poor position to understand the subject. I have read through everything you have written and I agree with you fully.

The parts I don't yet fully understand are what pka wrote using the interval notation, and what I mean by this is not how it is written, but how to use it in conjunction with a question asked!

So what do I mean, well let me try to explain.

Inequalities: The domain of g (x) = sqrt x is x greater than or equal to 0

That sounds easy doesn't it!

Well in my coursebook I could not decide which is the domain and which is the codomain?

No clear instruction was given, and they change their ideas so fast between domain, codomain, function of, rule, mapping etc, it all gets too much too quick and one cannot decide which part of the function belongs to what part?

Now looking at the inequality above, the domain of g (x) is sqrt of x, and because x is positive it must be equal to or greater than 0, so based on that I understand how the intervals play a part in conjunction with the inequalities.

So providing now I am on the right lines of understanding now I require some examples to read through and then try some practice examples myself, which this course does not provide worked examples so I can't see how things fit together?


But thanks for your input:smile: