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A function is much like a computer program. The function is a rule (e.g., an equation) that takes some inputs (numbers) and produces some output, another number. One kind of function is the type where we put one real number in and get one real number out. These are usually written, y = f(x), where x is the number we put in, y is the number we get out, and f(x) is the rule that says how to get from x to y. In this case, we say that y is a function of x.
A function is a special kind of relation where each input is only allowed to have one output. Any equation containing only the variables x and y describes a relation between x and y. If the equation allows only one value of y for each value of x, the equation describes y as a function of x.
Since the number we get out depends on what we put in, we call y the dependent variable and x the independent variable. One way to remember this is that the independent variable is the input, while the dependent variable is the output. If there was such a word as "outdependent", those clever mathematicians would have used it.
The set of numbers that can be given as inputs is called the domain of the function, while the set of numbers that are outputs of the function is called the range of the function.
The equation y = x2 defines y as a function of x. The variable x is independent and the variable y is dependent.
While x is usually the independent variable, we can also switch the letters around. If an equation allows only one value of x for each value of y, we could say the equation defines x as a function of y. Letters are just letters, after all. They are used to represent numbers. We could write y as a function of a purple elephant, if we wanted.
The equation x = y2 defines x as a function of y, because for each y there is only one possible value of x. If we take y as the independent variable and x as the dependent variable, we have a function x = f(y).
The same equation x = y2 does not define y as a function of x because for each value of x there are two possible values of y ( and ).
The equation x = y describes x as a function of y and also describes y as a function of x.
The equation x2 = y2 does not describe x as a function of y. How do we know for sure? Try a couple numbers in the equation. If y = 2 then x can be either 2 or -2. For any value of y there are two possible values of x. This equation doesn't describe y as a function of x either, because for any value of x there are two possible values of y. If x = 2 then y = ± 2.
What does it mean for something to be increasing or decreasing?
Albert is an industrious fellow. He decides to start his own lawn mowing business. When Albert gets paid for mowing his neighbor's lawn, he increases the amount of cash in his wallet. And when Albert decides go buy a pack of cotton candy bubble gum as a reward for his labors, he decrease that cash cache while increasing the tasty goodness in his mouth.
We say a function y = f(x) is strictly increasing if y gets bigger as x moves to the right. A strictly increasing function could look like one of these (or like many other things):
In symbols, if a < b then f(a) < f(b).
Sometimes, functions in real life are more complicated. We only deal with these functions in pieces called intervals.
We say a function y = f(x) is strictly increasing on the interval I if a < b implies f(a) < f(b) for all a, b in the interval I.
The function y = f(x) graphed below is increasing on the interval [x1, x2], but not on the whole real line:
We say a function y = f(x) is nondecreasing if y doesn't get smaller as x moves to the right. The function doesn't have to get bigger, but it's not allowed to get smaller.
In symbols, if a < b then f(a) ≤ f(b).
A function can also be nondecreasing on a particular interval.
We can say a function is increasing at a point if its slope at that point is positive.
This is different than increasing over an interval.
We recommend not using the word increasing all by itself. We instead say strictly increasing or nondecreasing to be clear.
All these definitions are for functions that, more-or-less, increase. There are similar definitions for functions that, more-or-less, go down.
A function y = f(x) is strictly decreasing if the function is always getting smaller.
In symbols, if a < b then f(a) > f(b).
A function y = f(x) is nonincreasing if the function never gets bigger. The function might stay the same for a bit, but it never increases.
In symbols, if a < b then f(a) ≥ f(b).
Be Careful: strictly increasing and strictly decreasing aren't opposites. It's possible to have a function that's neither:
Similarly, nondecreasing and nonincreasing aren't opposites. It's possible to have a function that's neither:
There's one other term used when talking about the increasing-ness or decreasing-ness of functions: monotonic. No, we aren't talking about Ben Stein. A function is monotonic if it's going only one way. Therefore, it is either going up or going down.
Bounded and Unbounded Functions: To Infinity and Beyond!
You may have met some bounded functions in algebra. Unfortunately, much like a friend of a friend of your ex-girlfriend or ex-boyfriend, you haven't seen them in a while and can't remember their name.
Here is a quick review of the ideas, first.
A function is bounded below if we can find some y-value K that the function never goes below.
In symbols, for all x we have K ≤ f(x).
A function is bounded above if we can find some y-value M that the function never gets above.
In symbols, for all x we have f(x) ≤ M.
If a function isn't bounded, we say it's unbounded.
A function which is unbounded above and increasing grows without bound:
If a function is unbounded below and decreasing, its magnitude grows without bound:
It's moving day on Calculus Street, and we need to decide which functions to pack up and take with us and which ones to leave behind for the next tenants.
The function f(x) = sin x is bounded above by 1 and below by -1.
We can put this function in our box and put it in the moving truck.
The function f(x) = ex not bounded above, but it is bounded below by 0. It doesn't ever equal 0, but we can still fit it in a box. We can't close the lid, so we'll have to put it at the top of the pile.
The function f(x) = x3 is not bounded above or below. We will leave this one here for the next tenants.
Be Careful: When a function has an upper and/or lower bound, the bound(s) are y-values, not x-values.
Unlike the typical college junior that shows up to their morning class wearing pajamas and their retainer, some functions care what they look like in the mirror. These functions, called even or odd functions, have some important properties we can take advantage of later.
Consider the y-axis to be a mirror. A function is even if it looks in the mirror and sees itself exactly as is. In other words, it looks the same when reflected across the y-axis.
For any value of x, the values f(x) and f(-x) must be the same.
In symbols, a function is even if f(x) = f(-x).
To check if a function is even we find f(x) and f(-x) and see if they're the same.
Alternatively, a function is odd if it looks in the mirror and sees itself standing upside-down, like the y-axis is a funhouse mirror.
For any value of x, f(-x) is the upside-down version of f(x). That is, f(x) and f(-x) are negatives of each other.
In symbols, a function is odd if f(-x) = -f(x).
If all the terms involve x raised to even exponents, the polynomial is even. Some examples are
f(x) = x2
f(x) = 6x4 – 7x2
f(x) = x8 + x6 + x22
If all the exponents are odd, the polynomial is odd. Some examples are
f(x) = x3
f(x) = x9 + 6x7
f(x) = x101+ x67
If some of the exponents are odd and some of the exponents are even, the polynomial is neither even nor odd.