Relations & Functions
Notes # Alpha 1
Relation:
a pairing of elements of one set with elements of a second set. This is usually expressed as a set of ordered pairs.
Domain:
the set of all abscissas (the first value in the ordered pair) of the ordered pairs.
Range:
the set of all ordinates (the second value in the ordered pair) of the ordered pairs.
Function:
a function is a relation in which each element of the domain is paired with exactly one element in the range.
Ex A: State the domain and range of each relation. Then state whether the relation is a function. Circle yes or no.
#1)
{(-1, 4), (3, -5), (6, 2)}
#2)
{(0, 5), (5, 7), (-3, -6), (0, 0)}
domain =
domain =
range =
range =
Function? Yes
No
Function? Yes
Note: If an element is repeated, you need to only write it once.
No
Ex B: Given that x is an integer, state the relation representing each of the following by listing a set of ordered pairs. Then state whether the relation is a function by circling yes or no. Note: Substitute #1) y = 2x â&#x20AC;&#x201C; 2 and 0 < x < 5 #2) y = 1.5x and 5 < x < 7 each number from the domain into the equation. Make sure you show all work.
Relation = Function?
Relation = Yes
No
Function?
Yes
No
Linear Relationships & Functions Page 1 of 5
Relations & Functions Function Notation: a function is commonly denoted by f. In function notation, the symbol f(x) is read “f of x”. f(x) = y, therefore the ordered pairs of a function are in the form (x, y) or (x, f(x)). Ex C: Given f(x) = |x2 – 11|, find each value.
#1)
f(0) =
#2)
f(-4) =
#3)
f(-
13 )
=
Note: You must simplify inside the absolute value before getting rid of abs value symbols.
Vertical Line Test: If every vertical line drawn on the graph of a relation passes through no more than one point on the graph, then the relation is a function. Ex D: Use a graphing calculator to determine whether each equation is a function. Note: When graphing, you must label all x- and y-int. You must also label vertices.
#1)
y = 5x2 + 1
Function?
Yes
#2)
No
y = (x – 3)½
Function?
Yes
No
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Relations & Functions
Notes # Alpha 1
Ex D: Use a graphing calculator to determine whether each equation is a function.
#3)
x2 + y2 = 49
Function?
Yes
No
Ex E: The symbol [x] means the greatest integer not greater than x. If f(x) = [x] + 6, find each value.
#1)
f(-4) =
#2)
Excluded values:
f (2.5) =
#3)
f(-6.3) =
Note: [x] basically means round down to the nearest integer.
real numbers when substituted in for “x” will give an imaginary number for “y” or be undefined.
Note: Excluded values usually come in two varieties. #1) values that make the denominator = 0. #2) values that make the radicand negative. Linear Relationships & Functions Page 3 of 5
Relations & Functions Ex F: Name all values of x that are not in the domain of the given function.
#1)
f ( x) =
x +8
Excluded values = #3)
f ( x) =
#2)
x 2 − 18 f ( x) = 32 − x 2
Excluded values = 4
x2 − 9
#4)
7 f ( x) = | 3 x | −5
Note: Since this has a radicand and a denominator, you have to make sure you do work for both. The excluded values are the union of the two answers.
Excluded values =
Note: To find excluded values, set the denominator = 0. Solve. The result is what x cannot be.
Note: When solving an absolute value you need to get the absolute value by itself. Then break the equation into two equations.
Excluded values = Linear Relationships & Functions Page 4 of 5
Relations & Functions
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