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1.1
Four ways to represent Functions
Definition of a Function
Theorem: Vertical Line TestA set of points in the xy - plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.
x
y
Not a function.
x
y
Function.
(a) For each x in the domain of f, there is exactly one image f(x) in the range; however, an element in the range can result from more than one x in the domain.
(b) f is the symbol that we use to denote the function. It is symbolic of the equation that we use to get from an x in the domain to f(x) in the range.
(c) If y = f(x), then x is called the independent variable or argument of f, and y is called the dependent variable or the value of f at x.
SummaryImportant Facts About Functions
Representing Functions
Representations of Functions
There are four possible ways to represent a function:
verbally (by a description in words) numerically (by a table of values) visually (by a graph) algebraically (by an explicit formula)
Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Verbally (with words)
orWith Diagrams:
Numerically: using Tables -
Visually: using Graphs -
Algebraically: using Formulas – There are several Categories of Functions:
1) Polynomial functions (nth degree, coefficient, up to n zeros or roots)2) Rational Functions: P(x)/Q(x) – Define domain.3) Algebraic functions: contain also roots. Ex: f(x)=Sqrt(2x^3-2) or
f(x)=x^2/3(x^3+1)4) Exponential functions: f(x)=b^x ; b: base, positive, real.5) Logarithmic functions: related to exponentials (inverse), logbx – b: base, positive
and not 1.• Most common: Exponential base e (2.718…) and inverse: Natural Log.
6) Trig. Functions and their inverses.
2
4(a)
2 3
xf x
x x
2(b) 9g x x
(c) 3 2h x x
2
The function is defined as
if < 0
2 if = 0
2 if > 0
(a) Find (-2), (0), and (3). (b) Determine the domain of .
(c) Graph .
f
x x
f x x
x x
f f f f
f
(d) Use the graph to find the range of .
(e) Is continuous on its domain?
f
f
Piecewise-defined Functions:Example:
The absolute value function is a piecewise defined function. Recall that the absolute value of a number a, denoted by | a |, is the distance from a to 0 on the real number line. Distances are always positive or 0, so we have
| a | 0 for every number a
For example,
| 3 | = 3 | – 3 | = 3 | 0 | = 0 | – 1 | = 1
| 3 – | = – 3
Important reminders about Absolute Value:
In general, we have
(Remember that if a is negative, then –a is positive.)
Absolute value function f (x) = |x|
x if x 0 |x| = –x if x < 0
Symmetry:
Even and Odd Functions
A function f is even if for every number x in its domain the number -x is also in its domain and
f(-x) = f(x)
A function f is odd if for every number x in its domain the number -x is also in its domain and
f(-x) = - f(x)
32h x x x
35 1g x x
23 24 xxxf
The inverse of f, denoted by f -1 , is a function such that f -1(f( x )) = x for every x in the domain of f and f(f -1(x))=x for every x in the domain of f -1:
Inverse Functions
TheoremThe graph of a function f and the graph of its inverse f-1 are symmetric with respect to the line y = x.
2 0 2 4 6
2
2
4
6
Increasing and Decreasing FunctionsThe graph shown below rises from A to B, falls from B to C, and rises again from C to D. The function f is said to be increasing on the interval [a, b], decreasing on [b, c], and increasing again on [c, d].
Figure 22
Example
In this graphthe function f (x) = x2 is decreasing on the interval (– , 0)
and increasing on the interval(0, ).
Figure 23