Question 1193790: Consider the function f(x)= 3/x^2 what is the domain and range of f(x) (interval notation)
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Answers:
Domain: (-infinity, 0) U (0, infinity)
Range: (0, infinity)
Explanation:
Division by zero is not allowed. We cannot have zero in the denominator. Something like 5/0 is undefined.
The denominator x^2 cannot have x = 0 for that reason. Any other x value is allowed.
The domain is the set of all real numbers x such that  .
Expressed as a disjoint set of two intervals, we can say: x < 0 or x > 0
The interval x < 0 is the same as -infinity < x < 0
The interval x > 0 or 0 < x is the same as 0 < x < infinity
We can shorten -infinity < x < 0 to the interval notation of (-infinity, 0). Use parenthesis to exclude each endpoint.
Similarly, 0 < x < infinity shortens to (0, infinity)
We then glue the two intervals to get the domain of (-infinity, 0) U (0, infinity)
The U refers to the union operator in set mathematics.
Imagine we had a number line stretching from -infinity to +infinity
Then poke a hole at 0 to remove it from the domain. Anything else is valid.
The range is the set of all possible y outputs
The numerator of y = 3/(x^2) is positive, and x^2 is always positive if x is a nonzero real number.
Therefore, 3/(x^2) overall is always positive for any nonzero real number x.
The range is y > 0 or 0 < y or 0 < y < infinity
This shortens to the interval notation of (0, infinity)
To visually verify those answers, I recommend using your favorite graphing tool like Desmos or GeoGebra to graph out y = 3/(x^2) to see why the domain and range the way they are.
Take note of the horizontal asymptote of y = 0 and the vertical asymptote of x = 0. They are perfectly overlapped with the x and y axis respectively.
The graph is here:
https://www.desmos.com/calculator/qj8c4oefpo
|
|
|
