__Functions__

__Function:__ correspondence between two sets that
assigns/matches the members of the first set to exactly one member of the second
set

__Set:__ group of numbers

**How Functions are Written**

*Functional Notation*

Written f(x) = x

Read "f of x is x" ~~~~~~~
f(x) DOES NOT mean multiplication!!!!

*Arrow Notation
*Written f:x --> x

Read "function f is assigned to x"

These do not necessarily have to equal x, they could be any expression. The letters do not always have to be the same either. You could have p(b) if wanted, but f(x), g(x), h(x), and the same of y are very common.

**What this means**

A function basically works as a numerical expression. If you have the function f(x) = x + 2, it means that the value of f(x) is 2 more than whatever the value of x is (either f(x) or x will be known). You simply replace every x in the expression with its value.

If x = 3, then f(3) = 3 + 2 = 5.

If x = 7 then f(7) = 7 + 2 = 9

The input into a function, in this case, x, is called the **domain**,
the replacement set for the variable.

The output, f(x), is called the **range**.

Each member of the domain may only map to one member of the range to be a
function, but many numbers in the domain may map to the same number in the
range. If any number in the domain maps to more than one member in the
range, it is a **relation**, but not a function. NOTE: f(x) = √x
is a function only because √ denotes positive square root. If the
expression was f(x) = ±√x, it would not be (for example, ±√4 = 2
or -2... the one number, 4, in the domain maps to two numbers, 2 and -2, in the
range).

If writing the domain, write D = {} (enclosing x | x = numbers between the brackets, separated by commas, or a description). Descriptions may include D = {x | x is a real number} or {x | x > 1}. x | x means "the set of all x such that x..." and then the conditions put on x. The same type of notation is used for the range, when it is R = {f(x) | f(x)...}. When writing the domain or range, write the numbers in ascending order.

Now, for some practice, evaluate these functions.

Find the range, given the domain D = {x | x = -3, -2, -1, 0, 1, 2, 3}

f(x) = x - 4

f(-3) = -3 - 4 = -7

f(-2) = -2 - 4 = -6

etc.

R = {f(x) | f(x) = -7, -6, -5, -4, -3, -2, -1}

f(x) = 2x + 5

R = {f(x) | f(x) = -1, 1, 3, 5, 7, 9, 11}

f(x) = x^{2
}f(-3) = (-3)^{2} = 9

f( 3) = 3^{2} = 9

etc.^{
}R = {f(x) | f(x) = 0, 1, 4, 9}

f(x) = x/6

R = {f(x) | f(x) = -1/2, -1/3, -1/6, 0, 1/6, 1/3, 1/2}

Sometimes, the output of a function (f(x)) is given, to find the input (x). If this happens, just substitute f(x)'s value for it and solve the equation left for x.

f(x) = 4x - 7, f(x) = 8

8 = 4x - 7

15 = 4x

x = 15/4

f(x) = x^{2} - 4, f(x) = 12

12 = x^{2} - 4

x^{2} = 16

x = ±4

A **piecewise-defined function**
is a function with a different formula used for different conditions on x.
For example:

f(x) = 3x - 1 for x ≥ 0

2x + 5 for x < 0

Find f(4), f(0), and f(-2).

f(4) = 3(4) - 1 because 4 ≥
0

f(4) = 12 - 1 = 11

f(0) = 3(0) - 1 because 0 ≥ 0

f(0) = 0 - 1 = -1

f(-2) = 2(-2) + 5 because -2 < 0

f(-1) = -4 + 5 = 1

When classifying functions, there is a way to fit them into one of three groups, even, odd, or neither.

An even function is a function for
which f(-x) = f(x).

An odd function is a function for which f(-x) = -f(x).

Is f(x) = |6x| even, odd, or
neither? Even, because f(-x) = |6(-x)| = |-6x|
which is the same as |6x|.

Is f(x) = 4x + 5 even, odd, or neither? Neither,
because f(-x) = -4x + 5 and -f(x) = -4x -5, not equal.

Is f(x) = 3x^{3} + x even, odd, or neither? Odd,
because f(-x) = 3(-x)^{3} + -x = -3x^{3} - x and -f(x) = -3x^{3}
- x.

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