is just the set of all numbers greater than or equal to 0.1.definition
A function is a rule for transforming an object into another object. The object you start with is called the input, and comes from some set called the domain. What you get back is called the output; it comes from some set called the codomain.
or example,f(x) = x 2. Since you didn’t say what the domain or codomain are, it’s assumed that they are both R, the set of all real numbers.
f is the transformation rule, while f(x) is the result of applying the transformation rule to the variable x. So it’s technically not correct to say “f(x) is a function”; it should be “f is a function.”
notice:
The transformed object doesn’t have to be different from the original one.
Now, let g(x) = x² with domain consisting only of numbers greater than or equal to 0.Since g and f have the same rule, but the domain of g is smaller than the domain of f, we say that g is formed by restricting the domain of f.
example:
suppose you have a dog called Junkster. Unfortunately, poor Junkster has indigestion. He eats something, then chews on it for a while and tries to digest it, fails, and hurls. Junkster has transformed the food into . . . something else altogether. We could let
j(x) = color of barf when Junkster eats x,
where the domain of j is the set of foods that Junkster will eat. The codomain is the set of all colors. For this to work, we have to be confident that whenever Junkster eats a taco, his barf is always the same color (say, red). If it’s sometimes red and sometimes green, that’s no good: a function must assign a unique output for each valid input.
range(值域）
The range is the set of all outputs that could possibly occur. You can think of the function working on transforming everything in the domain, one object at a time; the collection of transformed objects is the range. You might get duplicates, but that’s OK.
the range is actually a subset of the codomain. The codomain is a set of possible outputs, while the range is the set of actual outputs.
- If f(x) = x² with domain R and codomain R, the range is the set of nonnegative numbers.
- If g(x) = x² , where the domain of g is only the nonnegative numbers but the codomain is still all of R, the range will again be the set of nonnegative numbers.
- h(x) = number of legs x has,If h(x) is the number of legs the animal x has,then the range is all the possible numbers of legs that any animal can have. I can think of animals that have 0, 2, 4, 6, and 8 legs, as well as some creepy-crawlies with more legs. If you include individual animals which have lost one or more legs, you can also include 1, 3, 5, and 7 in the mix, as well as other possibilities. In any case, the range of this function isn’t so clear-cut; you probably have to be a biologist to know the real answer.
- Finally, if j(x) is the color of Junkster’s barf when he eats x, then the range consists of all possible barf-colors. I dread to think what these are, but probably bright blue isn’t among them.
2.Interval notation
[a, b] means the set of all x such that a ≤ x ≤ b. For example, [2, 5] is the set of all real numbers between 2 and 5, including 2 and 5. (It’s not just the set consisting of 2, 3, 4, and 5: don’t forget that there √ are loads of fractions and irrational numbers between 2 and 5, such as 5/2, 7, and π.) An interval such as [a, b] is called closed(闭区间).
(a, b) is the set of all numbers between a and b, not including a or b. So if x is in the interval (a, b), we know that a < x < b. The set (2, 5) includes all real numbers between 2 and 5, but not 2 or 5. An interval of the form (a, b) is called open(开区间).
You can mix and match: [a, b) consists of all numbers between a and b, including a but not b. And (a, b] includes b but not a. These intervals are closed at one end and open at the other. Sometimes such intervals are called half-open. An example is the set {x : 2 ≤ x < 5} from above, which can also be written as [2, 5).
There’s also the useful notation (a, ∞) for all the numbers greater than a not including a; [a, ∞) is the same thing but with a included. There are three
3.finding the domain 找到定义域
Most of the time, however, the domain is not provided. The basic convention is that the domain consists of as much of the set of real numbers as possible. For √ example, if k(x) = x, the domain can’t be all of R, since you can’t take the square root of a negative number. The domain must be [0, ∞), which is just the set of all numbers greater than or equal to 0.
Here's a list of the three most common possibilities of causing a screw-up.
1. The denominator of a fraction can’t be zero.
2. You can’t take the square root (or fourth root, sixth root, and so on) of a negative number.
3. You can’t take the logarithm of a negative number or of 0.
You might recall that tan(90 ◦ ) is also a problem, but this is really a special case of the first item above. You see,
so the reason tan(90 ◦ ) is undefined is really that a hidden denominator is zero. Here’s another example: if we try to define
then what is the domain of f? Well, for f(x) to make sense, here’s what needs to happen:
- We need to take the square root of (26−2x), so this quantity had better be nonnegative. That is, 26 − 2x ≥ 0. This can be rewritten as x ≤ 13.
- We also need to take the logarithm of (x + 8), so this quantity needs to be positive. (Notice the difference between logs and square roots: you can take the square root of 0, but you can’t take the log of 0.) Anyway, we need x + 8 > 0, so x > −8. So far, we know that −8 < x ≤ 13, so the domain is at most (−8, 13].
- The denominator can’t be 0; this means that (x−2) =6 0 and (x+19) =6 0. In other words, x =6 2 and x =6 −19. This last one isn’t a problem, since we already know that x lies in (−8, 13], so x can’t possibly be −19. We do have to exclude 2, though.
So we have found that the domain is the set (−8, 13] except for the number 2. This set could be written as (−8, 13]\{2}. Here the backslash means “not including.”