More on Functions and Their Graphs

Transcription

1 More on Functions and Teir Graps Difference Quotient ( + ) ( ) f a f a is known as te difference quotient and is used exclusively wit functions. Te objective to keep in mind is to factor te appearing in te denominator from te problem, as seen in te following examples. Example : Find te difference quotient of te function. a.) f(x) x + 3x + 4 Step. Find f(a) and f(a + ). f(a + ) (a + ) + 3(a + ) + 4 f(a + ) (a + a + ) + 3a f(a + ) a + 4a + + 3a f(a) a + 3a + 4 Step. Substitute into te difference quotient. ( ) ( ) f a+ f a a + 4a+ + 3a [a + 3a+ 4] a + 4a+ + 3a a 3a 4 4a (4a+ + 3) 4a+ + 3

2 Example (Continued): b.) f ( x) x + Step. Find f(a) and f(a+). ( ) f a+ + + ( a ) a + + ( ) f a ( a) + a + Step. Substitute into te difference quotient. f ( a+ ) f ( a) a+ + a+ a+ a+ + a+ a+ + a+ a+ + a+ a+ + a + a+ a+ a+ + a + a+ a+ a+ + a+ a+ + [ ] a + a+ a+ + a+ a a + a+ a+ + a + a+ a+ + a + a+ a+ + a + a+ a+ +

3 Piecewise Defined Functions: Te last type of function we will consider is called a piecewise defined function. It is not one of te basic functions we ave already looked at, so te graps will not follow any particular sape. Eac piecewise defined function is different, but some general guidelines can be used to grap tese functions. By piecewise defined, we mean tat te function is defined by a different expression for different values of te dependent variable. Te grap will ave different curves, or pieces coinciding wit te different ways tat te function is defined. Wit tese types of functions, it is necessary to look at eac piece of te function separately, ten grap. Example : Grap te following function: f ( x) 5 x x + if if if x 0 0 < x < 3 3 x Solution: We ave tree pieces of te grap to work wit. Step : First, we ave te segment were x 0 or (,0] in interval notation. x f ( x) Graping tis portion of te function, we ave: Notice, we ave a solid circle at x 0, since te condition for tis piece of te grap is for x to be less tan or equal to zero.

4 Example (Continued): Step : Graping te next portion of te grap were 0 < x < 3 is defined on te interval (0, 3), we ave te following table: x 0 3 f ( x) 0 3 x Graping tis portion of te function, we ave: Notice tat we ave open circles at te values were x 0 and x 3, because tis point is excluded from te grap based on te given conditions. Step 3: Te final portion of te grap is were x 3 is defined to be [3, ), so we can make te following table: x f ( x) x

5 Example (Continued): So we ave, We use a solid circle to sow tat te point were x 3 is included in tis portion of te grap. Step 4: Now we ave te tree pieces of te grap, but in order to correctly grap te wole function, we must grap all tree pieces. Tere are many different types of piecewise defined functions. Some ave patterns like stair steps, and oters ave less of a pattern (as in te above example). In eac case, you can follow te general steps just sown. Begin by breaking down te function into te separate pieces according to te conditions given for te independent variable (x in tis case). Ten you can grap eac piece. Remember to pay close attention to te signs <, >,, and. Tis will determine weter you ave an open or closed dot at te end points of te grap.

6 Increasing and Decreasing Functions: Functions tat are used to model real-life scenarios are usually not completely constant. Tere will be periods (or intervals) were tey increase or decrease. For example, if you set your cruise control at 70mp in your car, tere will be times wen te speed falls sligtly below or rises sligtly above 70mp (suc as wen you go up or down a ill). In tese instances, te function of te rate would be decreasing or increasing. Tis can be easily seen wen te function is graped. Wen te grap rises on an interval, te function is increasing. Wen te grap falls on an interval, te function is decreasing. Def: Increasing and Decreasing Functions: f is increasing on an interval if f ( x ) < f ( x ) wenever x < x f is decreasing on an interval if f ( x ) > f ( x ) wenever x < x

7 Example 3: For a particular small town in west Texas, te population was noted over a five year period of time. Te values are recorded on te following grap. (a) For wat period of time was te population increasing? (b) For wat period of time was te population decreasing? (c) For wat period of time was te population constant (i.e. no cange in population)? Solution: (a) We can see from te grap tat te population was increasing from 98 to 983. (b) We can see from te grap tat te population was decreasing from 980 to 98. (c) Te population was constant from 983 to 984. Relative Extrema Te relative extrema of a function are te points were a relative (local) maximum or minimum point exists in te open interval (a, b). Wen locating te relative extrema you will want to look at te critical numbers derived from te first derivative and any endpoints of te function. A relative maximum would be te igest point in te open interval (a, b). Terefore, te value of te function at te relative maximum point sould be greater tan te value of te function at all oter points in te same open interval. If c is a number located in te open interval of (a, b) and is included in te domain of te function f, ten a relative maximum exists at f(c) wen f ( x) f ( c) for all x in te open interval (a, b)

8 A relative minimum on te oter and would be te lowest point in te open interval (a, b). If c is a number located in te open interval of (a, b) and is included in te domain of te function f, ten a relative minimum exists at f(c) wen f ( x) f ( c) for all x in te open interval (a, b)

9 Even and Odd Functions: A function f is an even function if f ( x) f (x) for all x in te domain of f. For instance, te function f (x) x is even because f ( x) ( x) x f (x). If a function f is even, ten te grap of f is symmetric wit respect to te y-axis. Tis means tat if we reflect te grap of f in te y-axis we will obtain te same grap. A function f is an odd function if f ( x) f (x) for all x in te domain of f. For instance, te function f (x) x 3 is odd because f ( x) ( x) 3 x 3 f (x). Te grap of an odd function is symmetric about te origin. Tis means tat if we reflect te grap in te y-axis, and ten reflect it in te x-axis we will obtain te same grap.

10 Example 4: Determine weter te functions are even, odd, or neiter. (a) g (x) 3x 7 + 7x 3 + x (b) f (x) x 6 + 9x 00 (c) (x) 0x 3 x Solution (a): Step : First we will determine wat g ( x) is. We do tis by substituting x for eac x in g (x) 3x 7 + 7x 3 + x and simplifying. Terefore 7 3 ( ) 3( ) + 7( ) + ( ) g x x x x 7 3 3x 7x x ( 3x 7x x) ( ) g x Step : Since g ( x) g (x), te function g (x) 3x 7 + 7x 3 + x is not even. Step 3: Since g ( x) g (x), te function g (x) 3x 7 + 7x 3 + x is odd. Solution (b): Step : First we will determine wat f ( x) is. Substitute x for eac x in f (x) x 6 + 9x 00 and simplify. ( ) ( ) ( ) ( x) 6 f x x + 9 x 00 6 x + 9x 00 f Step : Since f ( x) f (x), te function f (x) x 6 + 9x 00 is even. Step 3: Since f (x) x 6 + 9x 00 is even, we do not need to test weter te function is odd. note: Te only function tat is bot even and odd is f (x) 0.

11 Example 4 (Continued): Solution (c): Step : Determine wat ( x) is. ( ) 0( ) ( ) 0x x 3 x x x 3 Step : Since ( x) (x), te function (x) 0x 3 x is not even. Step 3: Since ( x) (x), te function (x) 0x 3 x is not odd. Step 4: Terefore te function (x) 0x 3 x is neiter even nor odd. Example 5: Based on te graps of te functions, determine weter te functions are even, odd, or neiter. (a) (b) (c) Solution (a): We determine weter a grap is even, odd or neiter by cecking its symmetry. Te grap of (x) is symmetric to te y-axis, terefore (x) is an even function. Solution (b): Te grap of g (x) is neiter symmetric to te y-axis nor to te origin, terefore g (x) is neiter an even nor odd function. Solution (c): Te grap of f (x) is symmetric to te origin, terefore f (x) is an odd function.