Calculus Solver How to find the range in math
Range has two meanings. In a data set it is one number: the largest value minus the smallest. For a function it is the set of every output the function can produce, usually written as an interval. Which one you need depends on whether you are handed a list of numbers or a rule.
Range of a data set
Sort the numbers, or just find the two extremes, and subtract: range = maximum - minimum. For 4, 9, 2, 15, 7 the range is 15 - 2 = 13.
It is a single number describing spread, and it is fragile: one unusual value moves it completely. Add a 90 to that list and the range jumps to 88 while the rest of the data has not changed at all. That is why the interquartile range often replaces it.
Range of a function
Here the range is the set of outputs, so the question is which y values the function can actually reach. The reliable method is to work out the shape first and read the extremes off it.
For y = x² - 1 the square is never negative, so the smallest output is -1 at x = 0, and everything above is reachable: the range is [-1, ∞). For y = √x the output is never negative, so the range is [0, ∞). For y = 1/x the output is never zero, giving (-∞, 0) ∪ (0, ∞).
Domain and range are not the same
The domain is what you are allowed to put in, the range is what can come out. They are answers to different questions and often look nothing alike.
For y = √x the domain is [0, ∞) because negatives cannot go in, and the range is also [0, ∞) because negatives cannot come out. The match is a coincidence of that function, not a rule.
Range of some common functions
| Function | Range | Why |
|---|---|---|
| y = x² | [0, ∞) | A square is never negative |
| y = x² - 1 | [-1, ∞) | The lowest point sits at -1 |
| y = √x | [0, ∞) | The root symbol gives the non-negative value |
| y = 1/x | (-∞, 0) ∪ (0, ∞) | The output never reaches zero |
| y = sin x | [-1, 1] | The wave is bounded both ways |
Questions
What is the range of a set of numbers?
The largest value minus the smallest. For 3, 8, 1, 10 it is 10 - 1 = 9.
What is the difference between domain and range?
Domain is the set of allowed inputs, range is the set of possible outputs. Read domain off the x axis and range off the y axis.
Can a range be negative?
For a data set, no: subtracting the smallest from the largest can never give a negative. For a function, yes: the outputs themselves can be negative, as with y = -x².
More calculus reference
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