Download Chapter 3. Functions 3.1 What is a Function

Chapter 3. Functions
3.1 What is a Function
Definition of Function A function f is a rule (or a set of rules) that assigns
to each element x in a set A (called the domain of the function) exactly one
element, called f (x), also known as the value or the image of f at x, in a set
B (the co-domain of the function).
Examples: 1. The rule/algorithm that assigns each resident of the U.S. a
social security number is a function. (What are its domain and range?)
2. The rule that assigns to each person his or her age (in years) is a function.
3. The set {(1, 2), (2, 1), (3, 3)} is a function.
4. The set {(1, 2), (2, 1), (1, 3)} is not a function. (Why not?)
In the notation y = f (x), x is the independent variable, y is the dependent
variable (or the value/image of f at x) .
Note: The co-domain is not necessary the range of f. The range of f is the
set of all actual values of the function. The co-domain is the (possibly
larger) set where all the values (and perhaps some other elements that are not
values) of the function come from. An easy example to illustrate the
difference is the function f (x) = 2x, where x is any integer. Since twice of
any integer is another integer, the set of all integer Z can be considered as
the co-domain (it is also the domain, by the way). The range is, however,
only the subset of all even integers.
Piecewise Defined Functions
Example: The unit step function (or Heaviside function), defined by
0,
uc ( x) = 
1,
x<c
x≥c.
Example:
 3 x 2 − 2,

F ( x) = e 5 x + x,
 cos( 2 x),

x<4
4≤ x<9
x≥9
.
Finding the Domain of a Function
Some rules of thumb: Polynomial, exponential, sine and cosine functions
have as their domain the set of all real numbers. The domain of rational
functions and other trigonometric functions is the set of real numbers
EXCEPT the zero(s) of their denominators. In general, the domain of a
function is the set of all real numbers for which the function’s expression is
defined.
Example:
f ( x) =
Example:
f ( x) =
x2 − x − 6
−1
x2 + 4
3.2 Graphs of Functions
The graph of a function is the set of all ordered pairs (x, f (x)) for all x in
the domain of f. That is, the graph of f is the set of all points (x, y) such
that y = f (x).
Some Basic Functions’ Graphs
[See the document "Graphs You Should Know".]
The Vertical Line Test
A curve in the coordinate plane is the graph of a function if and only if no
vertical line intersects the curve more than once.
Example: A circle or ellipse is not the graph of any function
Example: The equation y = | x | defines a function, but the equation
x = | y | does not.
n
n
Example: Two families of power functions: y = x and y = 1 / x .
When n = an even positive integer:
When n = an odd positive integer:
Determine if an Equation Defines a Function (algebraically)
Example: Without using the vertical line test, can you tell whether each
equation describes a function?
3
(a) x = y + 1
2
2
(b) y = 4x + 1
3.3 Increasing and Decreasing Functions;
Average Rate of Change
Increasing and Decreasing Functions
f is increasing on an interval I if f (x1) < f (x2) whenever x1 < x2 in I.
f is decreasing on an interval I if f (x1) > f (x2) whenever x1 < x2 in I.
Example: The function f (x) = mx + b is increasing on the entire number
line if m > 0. It is decreasing on the entire number line if m < 0.
2
Example: The function f (x) = x is decreasing on the interval (−∞, 0), and
is increasing on the interval (0, ∞). It is neither increasing nor decreasing at
0.
Example: The function f (x) = 1/ x is decreasing on the intervals (−∞, 0)
and (0, ∞). (It is undefined at 0.)
Average Rate of Change
The average rate of change of the function y = f (x) between x = a and x =
b is
Avg. rate of change =
∆y f (b) − f (a)
=
.
∆x
b−a
Geometrically, the average rate of change is the slope of the secant line
between x = a and x = b on the graph of f , that is the line that passes
through the points (a, f (a)) and (b, f (b)).
Example: Find the average rate of change of the function
f ( x) =
x2 − 2 x + 2
3x + 1
(i) Between x = 0 and x = 2, and (ii) between x = 1 and x = 4.
Example: Every linear function, y = mx + b, has a constant average rate of
change, m, on any interval.
On any interval from x1 to x2, the average rate of change is
f (b) − f (a) (mx2 + b) − (mx1 + b) mx2 − mx1 m( x2 − x1 )
=
=
=
= m.
x2 − x1
x2 − x1
x2 − x1
x2 − x1