Download Math 11: Chapter 4 What is a Function? Def: A “Relation” is a

Math 11: Chapter 4
1. What is a Function?
a) Def: A “Relation” is a correspondence between 2 elements (a random set of order pairs or
coordinates).
b) Def: A “Function” is a rule of correspondence in which each first element is paired with one
and only one second element (there is no repetition of x-values in the set of coordinates that
represents the relation).
c) Def: The “Domain” of a relation is the set of all first elements (the set of all x-values).
d) Def: The “Range” of a relation is the set of all second elements (the set of all y-values).
e) Def: The “Pre-Image” of an expression is the input value (value to substitute) or x-value.
f) Def: The “Image” of an expression is the output value or y-value.
g) Notation: f ( x ) is defined as “the function of x”, as in the y-value you produce after an xvalue has been substituted into the equation.
2. The Graph of a Function. Key points-There is 2 methods for determining whether a relation is a
function. Either check the collection of points for any repetition of x-values or use the vertical line
test on the graph of the relation.
3. Properties of a Function. Key point-utilizing or transitioning into new notation of f ( x ) .
Ex: f (2)  5 is equivalent to the coordinate (2, 5).
a) Odd Functions:
i.
Odd function polynomials have only odd exponents.
ii.
Odd functions are symmetric about the origin (they have point or 180 rotational
symmetry).
b) Even Functions.
i.
Even function polynomials have only even numbered exponents.
ii.
Even functions are symmetric about the y-axis.
4. Composition of Functions (order specific substitution). Key points-when performing the composition
of functions either work from the inner most set of parenthesis or from right to left. Also, understand
that the notation f ( g ( x))  f g ( x) . Ex: Find f ( g (5)) ; if f ( x)  x 2 and g ( x)  x  4 . First
substitute 5 into g ( x ) , then take your result and substitute it into f ( x ) .
5. The Inverse of a Function.
a) Def: An “Inverse of a Relation” performs the following operation to each and every
coordinate: ( x, y )  ( y, x) .
b) Def: An “Inverse Function” is both one to one and passes the horizontal line test as well as the
vertical line test.
c) Def: In a “One to One” Function, every x-value has one and only one y-value, and likewise
every y-value has one and only one x-value (neither x or y repeats).
d) To Show Algebraically that 2 functions are inverses, perform the following compositions of
functions: f g ( x)  x and g f ( x)  x
e) To Show Graphically that 2 functions are inverses, graph both functions and the line y  x
to show symmetry over the line y  x .
f) To find an inverse, follow the following procedure:
i. Check to see if the original function is one-to-one.
ii. Switch x and y.
iii. Solve the new equation for y.
iv. Make sure to use the inverse notation of f 1 ( x)
v. Key point-Inverse Functions usually consist of opposite operations.