Functions Review Sheet
Function Facts:
A function has one unique y-value for each x-value. If something has two or more yvalues for a single x-value, then we call it a relation. In a mapping this means that each xvalue maps into only one y-value (note the x-values must be unique (no repeats):
Example:
Function
Not Functions (relations)
Domain and Range:
All possible x-values make up the domain of a relation or a function. The two things that
will restrict domain values are the following two hard fast rules:
1) We never can divide by zero (only occurs when I have x in the denominator)
2) We don’t let numbers underneath a square root become negative
If we are only given a graph, then look to see what values x can take on based on what’s
graphed (see example below).
Examples:
x² + 3x – 8
f(x) = --------------x-3
so x ≠ 3
g(x) = √x – 2
so x  2
domain of f(x) is {all real numbers, except x = 3} or {x | x ≠ 3}
domain of g(x) is {all real numbers 2 or larger} or {x | x  2}
All possible y-values make up the range of a relation or a function. We plug in the
possible x-values and determine the corresponding range values. If we have a graph then
we read the range off of possible y-values.
Examples:
Graph of k(x)
What is the range of the function
f(x) = ½x – 2 when the domain is
{2, 4, 6}?
Plug in x-values, solve for y
and those are the range values
{-1, 0, 2} in the example above
Domain { x | x is all real numbers}
Range { y |y ≥ 0} (no negatives!)
Functions Review Sheet
Looking at a graph to see if it’s a function:
When looking at a graph to see if it is a function we use the Vertical Line Test.
Vertical Line Test: If a vertical line touches a graph more than once, then it is not a
function (but a relation). The following examples are graphs that fail the vertical line test.
The following graphs are examples that pass the vertical line test
Ordered Pairs: In a set of ordered pairs (like the mappings we first looked at), for every xvalue there must be only one unique y-value for a function. In the example below:
A {(2, 3), (4, 1), (2, 1), (1, 5)}
B {(1, 4), (2, 3), (3, 2), (4, 3)}
C {(2, 3), (3, 2), (4, 4), (5, 2)}
D {(2, 3), (1, 4), (2, 3), (1, 5)}
answer A has the x-value 2 going to both 3 and 1 for y-values (so not a function)
answer B has all unique x-values (no repeats) so it is a function
answer C has all unique x-values (no repeats) so it is a function
answer D has the x-value 2 repeated twice, but both times it goes to 3 (one unique y-value,
so it is a function)