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Algebra (all content)

Course: Algebra (all content)>Unit 7

Lesson 23: Determining the domain of advanced functions (Algebra 2 level)

Domain of advanced functions

Sal covers many different kinds of functions and shows how to determine their domain. Created by Sal Khan.

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• HOW COME 1/O IS WRITTEN AS UN-DEFINED OR INFINITE? CAN SOMEONE PLZ EXPLAIN HOW 1/O IS UNDEFINED?????????????????
• Jennifer is correct, but here's another way to show that it's impossible to divide 1 by 0.
Let's say that there is an answer. Let's call that answer "q"
1/0 = q
Let's multiply both sides by zero
1 = 0*q
1 = 0
Ridiculous!

The answer can't be infinity either because infinity isn't a number and "defined" means the answer is a number.
• At Sal says that f(x)=x^2 has a domain of all real numbers. Sqrt(-1) or i is not a real number, but when plugged into f(x) produces a valid output. What's going on here?
• While it is necessary to constrain the domain of the function to avoid numbers that give undefined outputs, there is nothing that says one cannot further constrain the domain for other reasons.

For example, suppose you measured velocities and distances only after the first hour of travel. Your resulting equations might make mention that time has to be greater than or equal to 1 hour. That doesn't mean that the math of the equations wouldn't give you a number for t = 0.50 hours, but that your equation itself wasn't set up to include that time range, so the output (while a single number) might be incorrect.

So, when a domain is constrained, that only means you didn't intend for that equation to be used for the forbidden values -- for any number of reasons.
• In the last function, should not the domain be Z(set of integers) - {1} as f(x) is only defined for even and odd numbers?
• I would tend to agree with you. I looked on google to try and determine if decimals or fractions are ever considered even or odd. There is some disagreement. But, most sites seem to indicate the concept of even and odd numbers only applies to integers. I think your domain would be better than Sal's.
• well if 1/0 is defined, dividing by zero or nothing, basically not dividing at all would just still be one wouldn't it?
• You have to think about what division means. It's basically saying how many of a number added together does it take to make the other.

So how many zeros do you need to make a 1? It doesn't matter how many you have, you will never get a 1, so this number is undefined.
• At 1.37 It seems to me that the domain would include some imaginary numbers too, since i squared gives an answer of -1 and 3i squared gives an answer of -9. Am I wrong? Or do you just only think about real numbers when stating the domain?
• That's why we need to state the domain. In normal exercises, the domain must be a subset of the real numbers ((so no imaginary numbers)), but as you say, complex numbers are completely fine; they are just not expected as answers for "domain questions".

Now, many times the domain including complex numbers can make a huge difference, as when you say "but that cannot be true as the domain of the function f is the real numbers" at the end of a reduction to absurd proof. We could say that the domain is a "variable" part of a function, just as its definition, so for example
f(x) = sin x ((0 < x < pi / 2))
g(x) = sin x ((x belonging to Complex numbers))
are completely different functions, as their domains are different.
• Shouldnt be only X > 3? Because if X < -3 wouldnt be possible to have 0 as result?
• Sal says he forgets if colon or a line is used. Which is it?
• Actually, you could use either. Some texts use a colon, and others use a vertical bar. Other texts use a backward rounded epsilon, which I have always found confusing. So how you annotate "such that" is up to you (or perhaps your teacher).
• THIS QUESTION IS OFF TOPIC, DUE TO THE FACT THAT I AM UNABLE TO FIND INTERVAL NOTATION

A question that has been confusing me for some time now :

*If you were given a function (ex. -10 < X <= 10, for simplicity reasons), and asked for interval notation, how would you do so correctly?*

Using the example formula below, here's how I went about the problem :
*(The higher line represents the thick line you graph on paper)*
<----◙⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃○---->
-10 -6 -2 0 2 6 10

After doing so, the interval notation I believed it to be is : `[10, 10)

According to my teacher, I was wrong. IS she right?
• First, here's the section of Khan Academy on interval notation:

Now then, on to your question.

-10 < X <= 10
In interval notation, that would be:
(-10, 10]

The reason for the parentheses on the left is because X cannot equal "-10", since the original notation says that "-10 < X". When X cannot equal that thing, you use parentheses instead of brackets.

The reason for the bracket on the right is because X can equal 10. When X can equal that number, you use brackets.

As for the number line you drew, that would appear to be:
[-10, 10)
OR
[-6, 10)

it seems a little off, because spacing doesn't work so well on Khan Academy posts. So I can't tell whether the -10 or the -6 was supposed to be under that black circle on the left of the number line.

I should point out, however, that the number line and the example you gave above (-10 < X <= 10) are not the same thing. The number line is (-10 <= X < 10), not the other way around.

Anyways, I hope this helps. And if I misunderstood any part of your question, just comment on my answer and let me know.
• Would 0/0 still be undefined?
(1 vote)
• Check out Khan's video concerning "Dividing by Zero"
• Lets say a plumber costs C (t) = 35 + 25t dollars per hour for jobs up to four hours long, where t is the number of hours. Find the values C(t+0.5) -C(t)? Then figure out for what values of t your answer will be true. Note that domain of the function is {t:0≤t≤4}.

Any ideas? How can I answer a question in this form, it isn't a homework question..I'm just trying to understand how to solve it myself! Thanks in advance!
(1 vote)
• Assuming the job is not more than 4 hours in both cases then do it in steps:
a) Solve for C(t+0.5) = 35 + 25 (t+0.5) = 47.5 + 25t
b) Solve for C(t) = 35 + 25t
Now take the difference between them which is:
(47.5 + 25t) - (35 + 25t) = 12.5