Properties of Functions

Y = f(x) read y as a function of x

X values independent variables

Y values dependent variables

Domain of f all possible inputs (x values)

Range of f set of outputs (y values)

Properties of FunctionsEx: 1) f(x) = x2 and 2) f(x) = x2, x≥3

Even though these equations look the same, they are different because they have different domains

Answers:

1) Range: y≥0 or 0≤y≤∞

2) Range: y≥9 or 9≤y≤∞

To fully describe a function you must not only specify the rule that relates to the inputs and outputs, but you must also specify the domain-the set of allowable inputs.

Graphs of Functions

F(x) = 1/x

F(x) = x2

F(x) = x3

F(x) = x

Graphs of Functions

xF(x) = F(x) = 3 x

Functions

Values of x for which f(x) = 0 are x-coordinates of the points where the graph of f intersects the x-axis. These are also known as:

• Zeros of f;

• Roots of f(x); or

• X-intercepts of y = f(x)

Graph of x2 + y2 = 25Is this a function?

No, because this is a circle centered at the origin with a radius of 5. Points on the circle are (3, 4) (3, -4) (4, -3) (4, 3) (-3, 4) (-3, -4) (-4, 3) (-4, -3) – does not pass the vertical line test or the horizontal line test.

225 xy

Algebraically:

225 xy

Break the equation into the union of two semicircles and you get: 225 xy

Each of these equations define y as a function of x. Passes the vertical line test so it is a function but not the horizontal line test so it is not one to one.

Absolute value function|x| = x, if x≥0 and –x if x<0

Properties of Absolute values:

a) |-x| = |x|

b) |xy| = |x||y|

c) |x/y| = |x|/|y|

d) |x + y| ≤ |x| + |y|

Basically, this graph is found by unionizing the two parts of the equation |x| = x, if x≥0 and –x if x<0

By definition, √x denotes the positive square root of x. To denote the negative square root you must write -√x.

Example: positive square root of 16 is √16=4 and the negative square root is -√16 = -4

Equations such as √x2 = x are not always true since if x = -5, then

√x2 = √(-5)2 = √25 = 5 ≠ -5Correct statement is √x2 = |x|

This is an example of piecewise function in the sense the formula for f changes, depending on the value of x.

Sketch the graph of the function defined piecewise by the formula:

0, x ≤ -1f(x) = √(1 – x2), -1 < x < 1

x, x ≥ 1 (the points where the graph changes are called the breakpoints)

Natural Domain

If a real valued function of a real variable is defined in a formula and if NO domain is stated explicitly, then it is understood that the domain is R numbers for which formula gives real value.

Ex: Find the natural domains

1) f(x) = x3 2) f(x) = 1/x 3) f(x) = 1/(x-2)

4) f(x) = tan x 5) f(x) =

652 xx

Effects of Algebraic Operations on the Domain

f(x) = x2 – 9 x – 3

Natural domain is all reals except x = 3 (b/c division by 0 is impossible)

Factoring the numerator and canceling the common factor obtains:

f(x) = (x – 3)(x + 3) = x + 3 which is defined

x – 3 at x = 3

Effects of Algebraic Operations on the Domain (continued)

So, algebra has altered the natural domain.

The graph of x+3 is a line whereas the graph of f(x) = x2 – 9

x – 3

is still a line but there is a hole at x=3 since it is undefined there.

The algebraic cancellation eliminates the hole in the original graph. To preserve the domain, express it in simplified form as f(x) = x+3, x≠3

Even and Odd Functions

Important Symmetric Properties

Even Function where f(x) = f(-x)

Odd Function where f(-x) = -f(x)

Even and Odd Functions• Even functions are symmetrical around y-

axis

y = x2

• Odd functions are

symmetrical around

the origin y = x3

(Odd functions: Rotation of 180° about origin leaves graph unchanged.)

Even or Odd?

1) y = x

2) y = x4

3) y = x + 3

4) y = √x

5) y = x2 + x4

6) y = x2 – 5

7) y = x1/3

Concepts of Relations and Functions and How They are Represented

• Functions are used by mathematicians and scientists to describe relationships between variable quantities

• Play a central role in calculus and its applications

• Use paired data

Study Hours

Regents Score

3 80

5 90

2 75

6 80

7 90

1 50

2 65

7 85

1 40

7 100

                          

                

Tables and Scatter Plot

Old Faithful Eruptions Scatter Plot

Line graph – join the successive points

Histogram/Bar Graph

Functions

Tables, graphs, and equations:

Provide three methods for describing how one property depends on another

Tables - numerical Graphs - visual

Equations - algebraic

A relation is a function if:for each x there is one and only one y.

A relation is a one-to-one if also: for each y there is one and only one x.

In other words, a function is one-to-one on domain D if:

f a f b whenever a b

To be one-to-one, a function must pass the horizontal line test as well as the vertical line test.

31

2y x 21

2y x 2x y

one-to-one not one-to-one not a function

(also not one-to-one)

If a variable y depends on a variable x in such a way that each value of x determines exactly one value of y, then we say that y is a function of x.

A function f is a rule that associates a unique output with each input. If the input is denoted by x, then the output is denoted by f(x) (read “f of x”).

Functions are represented four basic ways:1) Numerically by tables2) Geometrically by graphs3) Algebraically by formulas4) Verbally

Curve fitting

Converting numerical representations of functions into algebraic formulas

Discrete vs Continuous DataDiscrete Data: Data that makes discrete jumps.

Data represented by scatter plots consisting of isolated points. Data that has a finite number of values and there is space on a number line between 2 possible values. Usually whole numbers.

Continuous Data: Data that has values that vary continuously over an interval. Data that is continuous and unbroken curves. Usually a physical measurement, can increase/decrease in minutely small values.

Classify each set of data as discrete or continuous.

1) The number of suitcases lost by an airline.

Discrete. The number of suitcases lost must be a whole number.

2) The height of corn plants.

Continuous. The height of corn plants can take on infinitely many values (any decimal is possible).

3) The number of ears of corn produced.

Discrete. The number of ears of corn must be a whole number.

Classify each set of data as discrete or continuous.

4) The number of green M&M's in a bag.Discrete. The number of green M&M's must be a

whole number. 5) The time it takes for a car battery to die.

Continuous. The amount of time can take on infinitely many values (any decimal is possible).

6) The production of tomatoes by weight.

Continuous. The weight of the tomatoes can take on infinitely many values (any decimal is possible).

Homework

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