Graph of Functions Chapter Graph of Functions. Rectangular Coordinate Sstem and Plotting points The Coordinate Plane Quadrant II Quadrant I (0,0) Quadrant III Quadrant IV Figure. The aes divide the plane into four regions, called quadrants; those are numbered as in Figure.. The point (, ) lies on quadrant I, the point (-4, -) lies on quadrant III and so on. Eamples. Plot the points (3, ), (-, -), (0, -3), (, 0), (-5, 0) and (0, ) (0,) (3,) (-5,0) (,0) (-,-) (0, -3) Figure.. Distance and Midpoint Formulas The Distance formula: Consider two points P(a, b) and Q(c, d). Then we have the following measures RUN = PR c a, and RISE = RQ d b. Using Pthagorean 0
Graph of Functions theorem we have the distance between P and Q is PR ( c a) ( d b ) Q P R Figure.3 The distance formula between two points A(, ) and B(, ) is given b AB ( ) ( ) RUN RISE. Find the distance between the points (5, 7) and (-3, 8) The distance d (5 3) (7 8) 65 The Midpoint Formula: The midpoint of the line segment from A(, ) and B(, ) is given b, 3. Find the midpoint of the line segment from (5, 4) to (3, 6) Solution: The midpoint is (4, 5).3 Straight Lines and Slopes The well-known general form of a straight line is a b c 0 and the standard form of a straight line is m b, also known as slope-intercept form. In the second form m is called the slope of the line. Remember that if a straight line passes through A(, ) and B(, ) the slope is RISE m RUN 4. Find the slope of the straight line which passes through the points (7, 4) and (, 3) RISE 4 3 Solution: The slope is m RUN 7 5
Graph of Functions 5. Find the slope of the straight line whose equation is given b 6 0 0 5 Solution: we write the given equation 6 0 0 as. 3 3 The slope is m. 3 Properties of slopes The slope of non-vertical lines is a number m that measures how steep are the lines. The following cases ma arise: When m 0, the straight line is horizontal When m, the straight line makes 45 degree angle with the ais in the positive direction Larger the absolute value of the slope steeper the straight lines For a vertical line the slope is undefined The straight lines with positive slope are increasing and make acute angles with ais in the positive direction The straight lines with negative slopes are decreasing and make obtuse angles with ais in the positive direction If m and m are the slopes of two given straight lines then the following cases ma arise When m m, the straight lines are parallel When m m, the straight lines are perpendicular 6. Plot the following straight lines and observe the slopes a) b) c).5 d) e) f).5 Solution: e) b) d) a) f) c) Figure.4 7. Show that the straight lines 0 3 and 0 5 0 are perpendicular to each other. Solution: Observe that for 0 3 the slope is m 0 and on the other hand the slope of 0 5 0 is m. We have m m 0 and the straight lines are perpendicular.
Graph of Functions 8. Show that the straight lines 0 3 and 5 50 5 0 are parallel to each other. Solution: The straight line 0 3 has slope m 0 and the straight line 5 50 5 0 has also the same slope m 0. The straight lines are parallel..4 Basics of Functions In everda life we encounter situations in which one item is associated with another. For instance, the growth of a plant ma depend on amount of fertilizer or the understanding of a problem ma associate with hours of stud. Relation and Function: A relation is a set of ordered pairs. A function is a relation in which no two ordered pairs have the same first coordinate. In the ordered pairs the set of all first coordinates constitutes the domain and the set of all second coordinates is the range of the relation. Formal definition of function A function is a rule that provides for ever single value of a corresponding unique value of. The domain of the function is the set of all permissible values of ; on the other hand the range of the function is the set of all possible values of. Vertical line test to determine a function A graph defines a function if an vertical line intersects the graph at no more than one point. a) Following are the graphs of functions Fig.5 Fig.6 Fig.7 b) Following are not the graphs of functions Fig.8 Fig.9 Fig.0.5 Graph of Functions 9. Plot the function f () 3
Graph of Functions Solution: This function is called an identit function. For ever single value of we have a unique value of equals. Thus following are the points on the graph of the function. - - 0 f () - - 0 Table. 0. Plot the function Fig. Solution: It is a vertical straight-line at units to the right from the origin. You tr to make a plot.. Plot the function 3 5 Solution: Let us consider the following points - - 0 f () - 5 8 Table. Fig.. Plot the quadratic function 4
Graph of Functions -3 - - 0 3 f () 8 3 0-0 3 8 Table.3.6 Transformation of Functions Fig.3 In this section ou will observe when the rule of a function is algebraicall changed in certain was to get a new function, then the graph of the new function can be obtained from the graph of the original function b simple geometric transformations. Below we discuss horizontal and vertical shifting. Vertical shift Consider the original function f ( ) and its graph then compare with two other functions g( ) f ( ) and h( ) f ( ). Look at the graphs below. And notice that the graph of g() is found from the graph of f () b shifting it verticall unit downward. On the other hand the graph of h( ) f ( ) is found from the graph of f () b shifting it verticall unit upward. h( ) 6 5 f () 4 3 g( ) -4-3 - - 4 - Fig.4 5
Graph of Functions Rule on vertical shift: The graph of g( ) f ( ) k is found b shifting the graph of f () verticall b k units. If k is positive then the shift is upward and if it is negative the shift is downward. Horizontal shift Consider the original function f ( ) and its graph then compare with two other functions g( ) ( ) f ( ) and h( ) ( ) f ( ). Look at the graphs below. And notice that the graph of g() is found from the graph of f () b shifting it horizontall unit to the right. On the other hand the graph of h() is found from the graph of f () b shifting it horizontall unit to the left. 6 h( ) ( ) 5 4 3 g( ) ( ) -4-3 - - 3 4 Fig.5 Rule on horizontal shift: The graph of g( ) f ( h ) is found b shifting the graph of f () horizontall b h units. If h is positive then the shift is to the left and if it is negative the shift is on right. Reflection The graph of g( ) f ( ) is found b reflecting the graph of f () along -ais. = f() = - f() Fig.6 6
Graph of Functions Stretch (Elongate or epand) and Shrink (Compress or contract) The following diagram is useful to remember the stretch and shrink of a graph b a known factor c, with the following values. We consider the cases f ( c ) for horizontal stretch ( H ST ) or horizontal shrink ( H SH ) and cf () for vertical stretch ( V ST ) or vertical shrink ( V SH ).. c c H ST H SH V SH V ST Rule on vertical stretch and shrink: The graph of g( ) cf ( ) is found b Shrinking verticall the graph of f () b a factor of c when c Stretching verticall the graph of f () b a factor of c when c Note: Remember that for g( ) cf ( ), when c is negative we have horizontal reflection or reflection along -ais. Rule on horizontal stretch and shrink: The graph of g( ) f ( c ) is found b Stretching horizontall the graph of f () b a factor of / c when c Shrinking horizontall the graph of f () b a factor of / c when c Note: Remember that for some cases of g( ) f ( c ), when c is negative we have vertical reflection or reflection along -ais. a) Compare the graphs of f ( ), g( ) (0.5 ) f (0.5 ). Observe that the graph of g( ) ( ) f ( ) and f ( ) is horizontall shrunk b a factor of c = / for the coordinates (divide coordinates b or multipl b /) to get the graph of g( ) ( ) f ( ), while the graph of f ( ) is horizontall stretched b a factor of c = 0.5 for the coordinates (divide coordinates b 0.5 or multipl b /0.5 = ) to get the graph of g( ) (0.5 ) f (0.5 ). g( ) (0.5 ) f ( ) g( ) ( ) Fig.7 7
Graph of Functions b) Compare the graphs of f ( ), h( ) 0.5 0.5 f ( ). Observe that the graph of g( ) f ( ) and f ( ) is verticall stretched b a factor of c = for the coordinates (multipl coordinate b ) to get the graph of g( ) ( ) f ( ). On the other hand the graph of f ( ) is verticall shrunk b a factor of c = 0.5 for the coordinates (multipl coordinate b 0.5) to get the graph of g( ) (0.5 ) f (0.5 ). g( ) ( ) f ( ) g( ) (0.5 ) Fig.8 Determining a function obtained from a series of transformations 4 3. Discuss the transformation of g ( ) compared to f( ) 4 Solution: We follow the following steps: Step. Get 4 f ( ), c 4, the graph is verticall stretched b a factor of 4 (multipl coordinate b 4) 4 Step. Get 4 f( ), from step, the graph moves unit horizontall to the right 4 Step 3. Get 4 f( ), from Step, the graph moves units upward. (,4) (,4) (, 5) f( ) (,) (-, -) (0, -4) F (-, -4) F F3 F4 Fig.9 We get second graph (F) multipling coordinate b 4, then move the graph horizontall b positive unit to get third graph (F3) and finall unit upward to get the fourth graph (F4), which is the final position. (0, -3) 8
Graph of Functions.7 Combination of Functions, Composite Functions Combinations of Functions In this section we will discuss about addition (sum), subtraction (difference), multiplication (product) and division (quotient) of functions. We also further discuss an important operation called composition of functions. Suppose f ( ), g( ) are two functions, then their sum is the function sa h () defined as h( ) f ( ) g( ) ( f g)( ) and the difference k( ) f ( ) g( ) ( f g)( ). In the same wa the other defined functions are m( ) f ( ) g( ) ( fg)( ) is a function from multiplication and n( ) f ( )/ g( ) ( f / g)( ), g( ) 0 is a function from division. The domain of all these combinations of functions is the common domain to both the functions. If D f is the domain of f( ), and D g is the domain of g ( ), then the common domain is Df D g. Note that for f ( )/ g( ), g( ) 0, the domain is Df D g when g ( ) 0. 4 4. For the functions f ( ) and g( ) functions and their domain determine the following a) f ( ) g( ) b) g( ) f ( ) c) g( ) f ( ) d) f( ), g ( ) 0 g ( ) Solution: a) 4 f ( ) g( ), the common domain is [0, ) b) 4 g( ) f ( ), the common domain is [0, ) c) 4 g( ) f ( ) ( ), the common domain is [0, ) d) f ( ) g ( ) 4, g ( ) 0, the common domain is (0, ) Composite function or composition of functions For two given functions f ( ), g( ), the compositions are defined as f ( ) g( ) f g f ( g( )) and g( ) f ( ) g f g( f ( )) The domain of f gis defined as i) is in g ( ) and ii) g ( ) is in f() 9
Graph of Functions 4 5. For the functions f ( ) and g( ) functions and their domain determine the following a) f ( ) g( ) b) g( ) f ( ) Solution: a) b) has domain all 0 f ( ) g( ) f ( g( )) f ( ) g( ) f ( ) g( f ( )) g( ) 4 4 has domain all such that 4 0which is true for all real values of. Thus the domain is the entire real line. 6. For the functions f( ) and g ( ) determine the following functions and their domain a) f ( ) g( ) b) g( ) f ( ) Solution: a) f ( ) g( ) f ( g( )) f The domain is which is domain for g ( ) and also for g ( ) should be defined on f() when g( ) 0. Thus the domain is all real values of, 0 b) g( ) f ( ) g( f ( )) g The domain is which is domain for f( ) and also for f() should be defined on g ( ) when f ( ) 0. Thus the domain is all real values of, 0.8 One-to-one and Inverse Functions One-to-one function If the inverse of an function is also a function then the function must be one-t0-one. For a one-to-one function f(), each in the domain has onl one image in the range. No in the range is the image of more than one in the domain. Thus if a function f() has an inverse it is one-to-one. And when the function f() is one-to-one then for an element and in the domain of f() with we will have f ( ) f ( ). A one-to-one function can be determined b horizontal line test. 30
Graph of Functions Horizontal line test If ever horizontal line intersects the graph of f () in at most one point, then the function is one-to-one. Fig.0 Fig. Fig. One-to-one function One-to-one function Not one-to-one 7. Graph the following functions and determine if it is a one-to-one function. 3 3 3 a) f ( ) 9 7 b) f ( ) 9 7 c) f ( ) 7 Solution: a) b) c) Fig.3 Fig.4 Fig.5 One-to-one function Not one-to-one one-to-one In Fig.5 the horizontal line 7 seems to have infinitel man values ver close to zero, still it considered to be one-to-one. Inverse Functions Two one-to-one functions f() and gare ( ) said to be inverses to each other if f ( ) g( ) g( ) f ( ) and we write f ( f ( )). If () range D. f ( ) g ( ). Note that f has domain D and range R then its inverse ghas ( ) domain R and If gis ( ) the inverse of f(), then the graph of gis ( ) the reflection of f() in the line, known as the line of smmetr. 8. We consider a one-to-one function f ( ) 3 and few points. Also suppose that gis ( ) the inverse of f(). We have following results 0 - f( ) f() 3 5 3 5 g ( ) 0 - Fig.6 g ( ) 3
Graph of Functions Determining the inverse of a one-to-one function f() Set f () Interchange the variables and so that the domain of inverse of f() is represented b numbers on the -ais. Solve for and replace b f ( ) 9. Find the inverse of f ( ) 3 f ( ) 3 Solution: f ( ) 3 3 3 3 3 f ( ) 3 0. Find the inverse of f( ) 3 Solution: We write, domain is all and range is all 3 Replace b and b to get Solve for : 3 3 ( ) 3 3 f ( ) has domain with all and range with all. 3 f ( ). Find the inverse of f ( ) Solution: We write, domain is all 0 and range is Replace b and b to get Solve for : f() f ( ) f ( ) 3