FUNCTIONS

GRAPHS OF FUNCTIONS; PIECEWISE DEFINED FUNCTIONS; ABSOLUTE VALUE FUNCTION; GREATEST INTEGER FUNCTION   OBJECTIVES: sketch the graph of a function; determine the domain and range of a function from its graph; and identify whether a relation is a function or not from its graph. • define piecewise defined functions; • evaluate piecewise defined functions; • define absolute value function; and • define greatest integer function

As we mentioned in our previous lesson, a function can be represented in different ways and one of which is through a graph or its geometric representation. We also mentioned that a function may be represented as the set of ordered pairs (x, y). That is plotting the set of ordered pairs as points on the rectangular coordinates system and joining them will determine a curve called the graph of the function. The graph of a function f consists of all points (x, y) whose coordinates satisfy y = f(x), for all x in the domain of f. The set of ordered pairs (x, y) may also be represented by (x, f(x)) since y = f(x).

Knowledge of the standard forms of the special curves discussed in Analytic Geometry such as lines and conic sections is very helpful in sketching the graph of a function. Functions other than these curves can be graphed by point-plotting. To facilitate the graphing of a function, the following steps are suggested: Choose suitable values of x from the domain of a function and Construct a table of function values y = f(x) from the given values of x. Plot these points (x, y) from the table. Connect the plotted points with a smooth curve.

EXAMPLE: A. Sketch the graph of the following functions and determine the domain and range.

SOLUTIONS: (9, 0) (0, 4) (-2, 3) (-3, 0) (3, 0) (0, 3) (-1, 1)

When the graph of a function is given, one can easily determine its domain and range. Geometrically, the domain and range of a function refer to all the x-coordinate and y-coordinate for which the curve passes, respectively. Recall that all relations are not functions. A function is one that has a unique value of the dependent variable for each value of the independent variable in its domain. Geometrically speaking, this means:

A relation f is said to be a function if and only if, in its graph, each vertical line cuts or touches the curve at no more than one point. This is called the vertical line test. Consider the relation defined as {(x, y)|x2 + y2 = 9}. When graphed, a circle is formed with center at (0, 0) having a radius of 3 units. It is not a function because for any x in the interval (-3, 3), two ordered pairs have x as their first element. For example, both (0, 3) and (0, -3) are elements of the relation. Using the vertical line test, a vertical line when drawn within –3  x  3 intersects the curve at two points. Refer to the figure below.

(0, 3) (3, 0) (-3, 0) (0, -3)

DEFINITION: PIECEWISE DEFINED FUNCTION Sometimes a function is defined by more than one rule or by different formulas. This function is called a piecewise define function. A piecewise defined function is defined by different formulas on different parts of its domain. Example: if if x<0 if

A. Evaluate the piecewise function at the indicated values. EXAMPLE: A. Evaluate the piecewise function at the indicated values. if x<0 f(-2), f(-1), f(0), f(1), f(2) if if if if f(-5), f(0), f(1), f(5)

EXAMPLE: B. Define g(x) = |x| as a piecewise defined function and evaluate g(-2), g(0) and g(2). Solution: From the definition of |x|,

EXAMPLE: Sketch the graph of the following functions and determine the domain and range.

DEFINITION: ABSOLUTE VALUE FUNCTION Recall that the absolute value or magnitude of a real number is defined by Properties of absolute value:

The graph of the function can be obtained by graphing the two parts of the equation separately. Combining the two parts produces the V-shaped graph. It may help to generate the graph of absolute value function by expressing the function without using absolute values. Example: Sketch the graph of the following functions and determine the domain and range.

DEFINITION: GREATEST INTEGER FUNCTION The greatest integer function is defined by greatest integer less than or equal to x Example: 1 3 1 -4 -1 1 2 1 2

Graph of greatest integer function. Sketch the graph of

y x o

EXERCISES: A. Given the following functions, determine the domain and range, and sketch the graph:

B.Compute the indicated values of the given functions. EXERCISES: B.Compute the indicated values of the given functions. a. b. c.

C. Define H(x) as a piecewise defined function and evaluate H(1), H(2), H(3), H(0) and H(-2) given by, H(x) = x - |x – 2|.