Explanation:
It depends on what you want to do. The topic of functions is easily a semester course in algebra, at least.
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A function is a relation that maps an input to a single output. Common representations are ...
Functions sometimes take multiple inputs to generate a given output.
Often, one of the first things you're concerned with is whether a given relation <em>is</em> a function. It <u><em>is not</em></u> a function if a given input maps to more than one output.
We say a relation <em>passes the vertical line test</em> when a vertical line through its graph cannot intersect the graph in more than one point. Such a relation <em>is a function</em>.
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When a function is written in equation form, it is often given a name (usually from the (early) middle of the alphabet. Common function names are f, g, h. Any name can be used.
When a function is defined by an equation, the variables that are inputs to the function are usually listed in parentheses after the function name:
f(x), g(a, b), h(m)
These variables show up in the function definition that follows the equal sign:
f(x) = 3x -4
g(a, b) = (1/2)a·b
h(m) = 1/(m^3 +3) +5
The listed variable is called the "argument" of the function.
This sort of form of an equation is sometimes called "functional form." That is, a dependent variable, such as y, can be defined by ...
y = 3x +4
or the same relation can be written in functional form as ...
f(x) = 3x +4
Sometimes students are confused by this notation, thinking that f(x) means the product of f and x. Yes it looks like that, but no, that's not what it means.
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One of the first things we like to do with functions is <em>evaluate</em> them. This means we put a particular value wherever the variable shows up.
If we want to evaluate the above f(x) for x=2, we put 2 (every)where x is:
f(x) = 3·x -4
f(2) = 3·2 -4 = 6 -4 = 2
We can evaluate the function for literals, also.
f(a) = 3a -4
f(x+h) = 3(x+h) -4 = 3x +3h -4 . . . here, h is a variable, not the function name
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We can add, subtract, multiply, divide functions, and we can compute functions of functions. The latter is called a "composition", and is signified by a centered circle between the function names.
<u>Add functions</u>: f(x) +h(x) = (3x +4) +(1/(x^3 +3) +5)
also written as (f+h)(x)
<u>Subtract functions</u>: f(x) -h(x) = (3x +4) -(1/(x^3 +3) +5)
also written as (f-h)(x)
<u>Multiply functions</u>: f(x)·h(x) = (3x +4)(1/(x^3 +3) +5)
also written as (f·h)(x) or (fh)(x)
<u>Divide functions</u>: h(x)/f(x) = (1/(x^3 +3) +5)/(3x +4)
also written as (h/f)(x)
<u>Function of a function (composition)</u>: f(h(x)) = f(1/(x^3 +3) +5) = 3(1/(x^3 +3) +5) +4
also written as (f∘h)(x) . . . . . the symbol ∘ is called a "ring operator". Sometimes a lower-case 'o' is used in plain text. It is not a period or dot or zero or degree symbol. Note the sequence of names means function f operates on the result of function h.
As with other function evaluations, the inner parentheses are evaluated first, and that result is then used as the argument of the outer function.
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Because a function name can stand for an algebraic expression of arbitrary complexity, we often use a function name to talk about the properties of expressions in general.
For example, if we want to reflect the graph of the function y = f(x) over the x-axis, we want to change the sign of every y-value. We can use function notation to write that idea as ...
y = -f(x) . . . . . f(x) reflected over the x-axis
The attached graph shows an example using the above function h(m).
