Let f:{x: IxI > 100} => R, f(x) = 1/x^2. State the range of f.
Let f:{x: IxI < 0.1} => R, f(x) = 1/x^2. State the range of f.
*R = real numbers
How do you find the two ranges? I know it sounds embarrassing but I am struggling to work out the range. Can you please show your working?
Answer to (a) is [0, 1/(10^4)]
Answer to (b) is [100, infinity]
the domain is (-∞,-100)U(100,∞)
Since f(x) = 1/x^2, the range is the same on both those intervals. So, R = (0,1/100^2)
if the domain is (-0.1,0.1) then f(0) is undefined. The range is thus (1/0.1^2,∞) since f(x) is symmetric about x=0.
To find the range of a function, we need to determine the set of all possible output values, or the set of "y" values that the function can take on.
For the function f(x) = 1/x^2, let's analyze the two given cases separately.
(a) In the first case, we have the function defined for x values such that |x| > 10. This means that x can take on any value greater than 10 or less than -10.
Let's consider the behavior of the function as x becomes larger and larger. As x approaches positive infinity, 1/x^2 approaches 0. Similarly, as x approaches negative infinity, 1/x^2 also approaches 0. Therefore, the function approaches 0 as x becomes extremely large (either positive or negative).
So, as x approaches infinity, f(x) (or 1/x^2) approaches 0. And since the function is always positive, the range of f in this case is [0, positive infinity).
However, we are given another condition, IxI > 100. It means that x values must be far enough from 0 so that their absolute value is greater than 100. This is important because we cannot divide by 0, so x = 0 is not in the domain of f.
When |x| > 100, we can be certain that 1/x^2 will always be less than or equal to 1/10,000 (1/(10^4)). This is the upper bound for the range of f(x) in this case.
Combining our findings, we can conclude that the range of f(x) for the first case is [0, 1/(10^4)].
(b) In the second case, we have the function defined for x values such that |x| < 0.1. This means x can take on any value between -0.1 and 0.1 (exclusive).
Again, let's analyze the behavior of f(x) as x becomes extremely small (close to 0). As x approaches 0, 1/x^2 becomes larger and larger because any number squared becomes larger as it approaches 0. So, as x approaches 0, f(x) approaches positive infinity.
Therefore, the range of f for this case is [positive infinity, infinity).
By comparing the two cases, we can conclude:
- The range of f when |x| > 10 is [0, 1/(10^4)].
- The range of f when |x| < 0.1 is [positive infinity, infinity).
I hope this explanation helps you better understand how to determine the range of a function!