How can I find the restrictions on the domain for this problem: f(x)=1/x. I have no idea how to find the range and domain for problems written like this.
Restrictions in the domain can occur when a value of x makes the function undefined. So what ever value of x makes the function undefined will be restricted.
Thank you for your explanations. Would that mean that my question would be all real numbers? Is there a specific way I could tell if the answer is all real numbers?
Hmmm... it may require some knowledge of how some functions work.
Take 1/x. You can plug any value into x and it will be defined unless x is 0. You cannot divide something by 0. 0 no matter how many times will not be equal to 1.
Another common one is sqrt(x). It's no longer a real number if x < 0. This is true for x^(1/n) where n is even.
Then there are logs and ln. You cannot take the logs and ln of 0 and negatives since you will get something unreal.
Okay, then the domain would be x> 0 but I'm not sure about the range. Is the domain right?
For 1/x. Try substitution a negative value into x. Will you get a real number? Yes. Only when x=0 will get you an undefined. So the domain is all real numbers but 0.
The range is all values that you can get when you plug in the domain. Err badly phrased..
Anyways, for 1/x you substitute in a large, large value such as 10 to the 93874372 power >.<. It will get a number very close to 0 but not quite. So you can never get a zero for 1/x no matter what value of x you substitute in as f(x) will only get closer and closer to 0 but never actually be 0.
However this does not mean the range is from 0 to 1 though. If you put a number less than 1 but greater than 0 into 1/x, f(x) will become greater then 1. Say you put .5 as x you will get 2. Substitute .05 you get 20, .005 you get 200. You should notice that as x becomes smaller and smaller, f(x) will become bigger and bigger. You can make x be infinitesimally small and f(x) will become infinitely large. Something similar will happen for values of x negative.
So the range will be all real numbers but 0 since you cannot ever get a 0 for f(x).
Okay, thank you very much.
I'm not sure what you meant by my question being badly phrased, but thank you for your response anyway.
You're welcome, I hope I was able to properly clarify some stuff up for you.
I meant the way I explained it. It sounded awkward >.>. Sorry for being err.. wishy washy? >.< My vocab isn't very.. large haha.
To find the restrictions on the domain of the function f(x) = 1/x, we need to consider which values of x are not allowed in order to avoid mathematical errors such as division by zero.
In this case, the function f(x) = 1/x involves division by x. Division by zero is undefined in mathematics, so we need to find the values of x that make the denominator (x) equal to zero.
To do this, we set the denominator equal to zero and solve for x:
x = 0
Therefore, zero is not allowed in the denominator, and the value x = 0 is the restriction on the domain of the function f(x) = 1/x.
In conclusion, the domain of the function f(x) = 1/x is all real numbers except zero, or in mathematical notation:
Domain: (-ā, 0) U (0, +ā)