0.1 Functions and Their Graphs

Real Numbers

 • A set is a collection of objects.• The real numbers represent the set of numbers

that can be represented as decimals.• We distinguish two different types of decimal

numberso A rational number is a decimal number that may can

be written as a finite or infinite repeating decimal, such as or = 2.333…2

Real Numbers (2)

An irrational number is a decimal number that does not repeat and does not terminate, such as = -1.414214… and = 3.14159…

• The real number set is represented geometrically by the real number line

Real Numbers (3)

• We use inequalities to compare real numbers x is less than y

x is less than or equal to yx is greater than yx is greater than or equal to y

yx yx yx yx

• We often use a double inequality such as

as shorthand for a pair of inequalities

and

When we use double inequalities, the positions of a, b, and c must be written as they would appear on the real number line if read from right to left or left to right.

Real Numbers (4)

cba

ba

cb

• Inequalities can be expressed geometrically or by using interval notation.

Real Numbers (5)

Real Numbers (6)

Real Numbers (7)

• The symbols

and

do not represent actual numbers, but indicate that the corresponding line segments extend infinitely far to the left or right.

Functions

• A function of a variable x is a rule f that assigns to each value of x a unique number f(x) (read ”f of x”), called the value of the function at x.

• x is called the independent variable.• The set of values that the independent variable is

allowed to assume is called the domain of the function. A function’s domain might be explicitly specified as part of its definition or might be understood from its context.

• The range of a function is the set of values that the function assumes.

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• Examples of functions:f(x) = 3x –1

f(x) = 9x3 + 7x2 – 8

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• Function Example: If we let f be the function with the domain of all real numbers x, and defined by the formula f(x) = 4x2 – 2x + 6, we can find the corresponding value in the range of f for a given value of x by substitution.

Find f(2): f(2) = 4(2)2 – 2(2) + 6 = 4(4) – 4 + 6 = 16 – 4 + 6 = 18

Find f(-3): f(-3) = 4(-3)2 – 2(-3) + 6 = 4(9) + 6 + 6 = 36 + 6 + 6 = 48

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• Function Example: Let f be the function with the domain of all real numbers x, and defined by the formula f(x) = (4 – x)/(x2 + 7).

Find f(h): Find f(h+2): f(h+2) =

7

4)(

2

h

hhf

7)2(

)2(42

h

h

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• When the domain of a function is not explicitly specified, it is understood that the domain of the function is all values for which the defining formula makes sense.The domain of f for f(x) = 2x is all real numbers.The domain of f for f(x) = 1/x is all real numbers except x = 0.The domain of f for is all real numbers greater than or equal to 0.

xxf )(

Graphs of Functions• We can express a function geometrically by

expressing it as a graph in a rectangular xy-coordinate system.

• Given any x in the domain of a function f, we can plot the point (x, f(x)). This is the point in the xy-plane whose y-coordinate is the value of the function at x.

• It is possible to approximate the graph of the function f(x) by plotting the points (x,f(x)) for a representative set of values of x and joining them by a smooth curve.

• We often use a tabular construct to capture the points we wish to plot.

Graphs of Functions (2)

Graphs of Functions (3)

Graphs of Functions (4)• To every x in its domain, a function assigns one and

only one value of y. That value is precisely f(x). This means:

The variable y is called the dependent variable because its value depends on the value of the independent variable x.

– For every curve that is the graph of a function, there is a unique y such that (x,y) is a point on the curve.Not every geometric curve is the graph of a function.

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• Vertical Line Test – A curve in the xy-plane is the graph of a function if and only if no vertical line can be drawn that will intersect the curve at more than one point.

• Which curves are the graphs of functions?

Three Views of a Function

• We how have three ways to describe a function• By giving the formula f(x) = … and defining the

domain of the independent variable (x). A function specified in terms of a formula is said to be defined analytically.

• By drawing the functions graph. Such a function is said to be defined graphically.

• By providing a table of function values (x and f(x)). This method is said to describe a function numerically.

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Graphs of Equations

• The equations arising in connection with functions all have the form: y = [an expression in x]

• Note that not all equations connecting x and y are functions.

• A graph of an equation can be plotted the same as the graph of a function. However, the resulting graph will only pass the Vertical Line Test if the equation represents a function.

Graphs of Equations (2)