8.2 Graphs of FunctionsThe graph of a function consists of all the points whose ordered pairs satisfy the function. For example, the graph of f(x) = 3x + 1 contains the point (2, 7) because 7=3 ∙2+1.

Not every graph is the graph of a function. The definition of a function specifies that no value of x can be paired with two or more values of y. Consequently, if a graph contains two or more points with the same x, the graph cannot represent a function.

Points having the same x lie above one another. They are points that lie on a vertical line.

Vertical Line Test for FunctionsIf any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.

See Example 1 on page 562.---------------------------------------------------------------------------------------------Information From a Graph

1) Use the Vertical Line Test to determine if the graph represents a function.

2) Determine the end-behavior. A closed dot indicates the graph does not extend beyond this

point and the point belongs to the graph. An open dot indicates the graph does not extend beyond this

point and the point does not belong to the graph. An arrowhead indicates that the graph extends infinitely in the

direction the arrow points.3) Find the function value for given value of x.4) Find the value of x that yields a given function value.5) Describe the general behavior of the function (increasing, decreasing,

etc.).

See Example 2 and Figures 8.8 – 8.11 on pages 563 – 564.

6) Determine the domain and range.

The domain of a function is the set of all real numbers for the independent variable, x. In the rectangular coordinate system, the x–values lie on the horizontal axis. Thus, to determine the domain of a function you look at the extent of the graph horizontally, that is, left–and–right. The range of a function is the set of all real numbers for the dependent variable, y. In the rectangular coordinate system, the y–values lie on the vertical axis. Thus, to determine the range of a function you look at the extent of the graph vertically, that is, up–and–down.

See Figure 8.12 on page 566.

See Example 4 on pages 567 – 568.

The domain and range are sets of numbers that can be specified using Set–Builder Notation or Interval Notation.

Review of Interval NotationInterval notation consists of two numbers or symbols representing the largest and smallest values (upper and lower bounds) in a continuous set of real numbers. If the set has no largest value, use the (positive infinity) symbol. If the set has no smallest value, use the (negative infinity) symbol.

If an end value is not included in the set, a parenthesis is used. If an end value is included in the set, a square bracket is used. A parenthesis is always used with positive and negative infinity.

All real numbers less than 4. (−∞, 4 ){ x|x<4 }

All real numbers less than or equal to 2. (−∞, 2 ]{ x|x≤ 2}

All real numbers greater than 7. (7 , ∞ )

{ x|x>7 } All real numbers greater than or equal to 1. [ 1 , ∞)

{ x|x≥ 1}

All real numbers between 5 and 11. (5,11)

{ x|5<x<11}

All real numbers between 3 and 8, inclusive. [ 3,8 ]{ x|3≤ x≤ 8 }

All real numbers. (−∞,∞)

The following table is Table 8.1 from page 566 of your textbook. It includes Graphing on a Number Line

Expressing a Continuous Set of Real Numbers Using Set Builder Notation, Interval Notation and Graphed on a Number Line

Set Builder Notation Interval Notation Graphed on a Number Line

{ x|a<x<b } (a , b) ------------|------------------|------------ a b

{ x|a ≤ x ≤b } [a ,b] ------------|------------------|------------ a b

{ x|a ≤ x<b } ¿ ------------|------------------|------------ a b

{x∨a<x≤ b } ¿ ------------|------------------|------------ a b

{ x|x>a } (a , ∞ ) ------------|------------------------------ a

{ x|x≥ a } ¿ ------------|------------------------------ a

{ x|x<b } (−∞,b) ------------------------------|------------ b

{ x|x≤ b } ¿ ------------------------------|------------ b

{ x|x is arealnumber } (−∞, ∞) ---------------------------------------------