Graphs and Functions(Review)

MATH 207

Distance Formula

Example: Find distance between (-1,4) and (-4,-2).

Answer: 6.71

Midpoint Formula

Example: Find the midpoint from P1(-5,5) to P2(-3,1).

Answer: (-4,3)

The standard form of an equation of a circle with radius r and center (h, k) is:

The Unit Circle equation is:

x

y

(h, k)

r(x, y)

Equations in two variables – Example: Circle

Equations

222 rkyhx

222 ryx

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.

2For the function defined by 3 2 , evaluate:f f x x x

(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

2

4(a)

2 3

xf x

x x

2(b) 9g x x

(c) 3 2h x x

Properties of Functions:

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)

Determine whether each graph given is an even function, an odd function, or a function that is neither even nor odd.

32h x x x

35 1g x x

23 24 xxxf

Where is the function increasing?

Where is the function decreasing?

Where is the function constant?

Local Maxima and Minima

Local Max

Local Min

Average rate of change of a Function

21Find the average rate of change of :

2f x x

From 0 to 1

Library of Functions (Famous Functions)

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:

Application problem:

Graphing Functions:

2

2

2

Use the graph of to obtain the graph of the following:

(a) 2

(b) 2

f x x

g x x

h x x

2Graph the function 2 3f x x

.

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

Theorem

The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.

f 1

2 0 2 4 6

2

2

4

6 f

f 1

y = x

(2, 0)

(0, 2)