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L2 graphs piecewise, absolute,and greatest integer
- 1. FUNCTIONS
- 2. 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
- 4. 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.
- 6. SOLUTIONS: (-3, 0) (3, 0) (0, 3) (-2, 3) (9, 0) (0, 4) (-1, 1) 2 )(.1 xxf = xxF −= 9)(.2 4)(.3 2 += xxG 1 23 )(.4 2 + ++ = x xx xh 2 9)(.5 xxh −= 23)(.6 ++= xxg ( ) [ )+∞ +∞∞− ,0: ,: R D ( ) [ )+∞ +∞− ,0: 9,: R D ( ) [ )+∞+ +∞∞− ,4: ,: R D ( ) ( ) 1,: 1,: ++∞∞− −+∞∞− exceptR exceptD [ ] [ ]3,0:R 3,3:D + +− ( ) [ )+∞+ +∞∞− ,3: ,: R D
- 7. 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:
- 8. 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. 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.
- 9. (0, 3) (3, 0)(-3, 0) (0, -3)
- 10. DEFINITION: PIECEWISE DEFINED FUNCTION if x<0 + = 1x x )x(f.1 2 0x ≥if A piecewise defined function is defined by different formulas on different parts of its domain. Example: − − − = 3x x9 x )x(f.2 2 0x ≤ 1x > 3x0 ≤< if if if Sometimes a function is defined by more than one rule or by different formulas. This function is called a piecewise define function.
- 11. if x<0 f(-2), f(-1), f(0), f(1), f(2) A. Evaluate the piecewise function at the indicated values. + = 1x x )x(f.1 2 0x ≥if − += 2 )2x( 1x x3 )x(f.2 f(-5), f(0), f(1), f(5) 0x <if if if 2x0 ≤≤ 2x > EXAMPLE:
- 12. B. Define g(x) = |x| as a piecewise defined function and evaluate g(-2), g(0) and g(2). EXAMPLE: Solution: From the definition of |x|, < ≥ − = 0x 0x if if x x )x(g 2)2(g 0)0(g 2)2()2(g Therefore = = =−−=−
- 13. Sketch the graph of the following functions and determine the domain and range. EXAMPLE: − = − = 2 23 )(.2 3 2 4 )(.1 x x xf xg if if if 1 21 2 −< <≤− ≥ x x x if if 1 1 ≥ < x x >+ ≤ = 112 1 )(.3 2 xifx xifx xf
- 14. DEFINITION: ABSOLUTE VALUE FUNCTION Recall that the absolute value or magnitude of a real number is defined by Properties of absolute value: <− ≥ = 0, 0, xifx xifx x yineaqualittriangleThebaba.4 valuesabsolutetheofratiotheisratioaofvalueabsoluteThe0b, b a b a .3 valuesabsolutetheofproducttheisproductaofvalueabsoluteThebaab.2 valueabsolutesamethehavenegativeitsandnumberAaa.1 +≤+ ≠= = =−
- 15. 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. xxf =)( <− ≥ = 0if, 0if, xx xx y Example: Sketch the graph of the following functions and determine the domain and range. 5x23)x(f.2 1x3x)x(f.1 ++= ++=
- 16. DEFINITION: GREATEST INTEGER FUNCTION greatest integer less than or equal to x The greatest integer function is defined by =x Example: =0 =1.0 =3.0 =9.0 =1 =1.1 =2.1 =9.1 =2 =1.2 =4.3 =− 4.3 =− 9.0 0 0 0 0 1 1 1 1 2 2 3 -4 -1
- 17. Graph of greatest integer function. xy =Sketch the graph of x xy = 1x2 −<≤− 0x1 <≤− 1x0 <≤ 2x1 <≤ 3x2 <≤ 2− 1− 0 1 2
- 19. EXERCISES: 22 2 2 4:.6 1 12 :.5 3:.4 21:.3 1:.2 34:.1 xyh x xx yg xyh xyG xyF xyH += − +− = += −= += += ( )( ) ( )( )312 943 .10 4:.9 23 211 13 :.8 312 31 :.7 2 22 +−+ −−+ = −= ≥ <<− ≤− = ≥+ <− = xxx xxx y xyG xif xif xif yf xifx xifx yF A. Given the following functions, determine the domain and range, and sketch the graph:
- 20. EXERCISES: B.Compute the indicated values of the given functions. 4t 4t4 4t if if if t 1t 3 )x(f > ≤≤− −< += )16(andf),4(f),6(f −− a. < ≤≤− −< − − = x2 2x2 2x if if if 3 1 4 )x(hb. c. )2(hand),e(h, 2 h),2(h),3(h 2 π −− −= −≠ − − = 3x 3x if if 2 4x )x(F 2 −− 3 2 Fand),3(F),0(F),4(F
- 21. 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|.