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