Objective: Identif functions and use correct notation to evaluate functions at numerical and variable values. A relationship is a matching of elements between two sets with the first set called the domain and the second set called the range. The following eamples show several was that relationships can be given. Eample. Suppose that the domain is five people in a classroom, {Ale, Bill, Chris, Diane, Eric} and the range is the colors, {red, blue, ellow}. Suppose the relationship shows each person s favorite color from those three choices. This can be shown with a mapping. Domain Ale Bill Chris Diane Eric Range Red Blue Yellow Eample. Another wa to epress a relationship is to give a set of ordered pairs. The set {(Ale, Red), (Bill, Blue), (Chris, Red), (Diane, Yellow), (Eric, Blue)} gives the eact same information as the mapping above. In a relationship, some values might be matched more than once. Note that in the above eample that each person is matched with just one color but some colors are used more than once. Eample 3. Suppose the following mapping uses the same domain and range as the first two eamples but now is showing the information about which colors each person is wearing. a) Who is wearing Red? (Answer: Ale) b) Which of those colors is Bill wearing? (Answer: Yellow) c) Who is wearing Yellow? (Answer: Bill and Diane) Notice that Ale is wearing both red and blue clothing. Chris is not wearing an of the colors listed in the range. Domain Ale Bill Chris Diane Eric Range Red Blue Yellow Page 4
Another wa to epress a relationship is with an equation. Normall denotes the range and the domain. Eample 4. If we have the equation 3 then we have a relationship where the domain and range are both the set of all real numbers. Here we can find the matching for an value of. For eample, if then 3(). This can be epressed as the ordered pair (,). Other ordered pairs that satisf the equation include (,5) and (7,3). Eample 5. If we have the equation + = then again we have a relationship. Here note that for a given value of that there might be more than one matching value of. For eample, if, then or. Both make the equation true. Another wa to epress a relationship is with a graph. Eample 6. This is the graph of the equation 3 from Eample 4. The graph shows the combination of and values which make the equation true. 5 5-4 -3 - - 3 4 - Eample 7. This is the graph of the equation from Eample 5. The graph shows the combination of and values which make the equation true..5.5 -.5 - -.5.5.5 -.5 - -.5 Page 4
Eample 8. A graph does not need to come from a nice equation. In the following eample suppose that the vertical value gives the temperature in degrees Celsius and the horizontal value gives the time in hours after midnight on a certain da. It visuall shows how the values are related. For eample we can see that at 8 a.m. the temperature was 5 degrees Celsius. The temperature was degrees Celsius at two different times: once at a.m. and again at 6 a.m. 6 4 8 6 4 3 4 5 6 7 8 9 DEFINITION OF A FUNCTION There is a special classification of relationships known as functions. Functions are relationships in which each value of the domain corresponds with eactl one value of the range. Generall is the variable that we put into an equation to evaluate and find. For this reason is considered an input variable and is considered an output variable. This means the definition of a function, in terms of equations in and could be stated as: there is eactl one value corresponding with an value. The bo below summarizes this definition and vocabular. DEFINITION OF A FUNCTION A relationship represents a function if each input value is matched with eactl one output value. The domain is the set of all input values. The range is the set of all output values. Page 43
Let s revisit some of the eamples above to determine whether each relationship represents a function. Eample : Each name is matched with eactl one favorite color so the relationship is a function. Eample : This eample shows the same relationship as Eample but as a set of ordered pairs rather than as a diagram. Notice that no two ordered pairs have the same first component but different second component. So as we alread know, the relationship in Eample is a function. Eample 3: Ale is wearing two different colors so one input (Ale) is matched with more than one output (red and blue). The relationship is not a function. Eample 5: We saw if, then or. Since we know of at least one value of having more than one matching value, the relationship is not a function. Now we will consider some new eamples and determine if the relationship represents a function. Eample 9. Which of the following sets of ordered pairs represents a function? a) {(, a), (, b), (3, c), (4, a )} is a function, since there are no two pairs with the same first component. The domain is the set {,, 3, 4} and the range is { abc.,, } b) {(, a), (, b), (, c), (4, a )} is not a function, since there are two pairs with the same first component but different second components: (, a ) and (, c ). Eample. Which of the following mapping diagrams represents a function? a) b) Domain Range Domain Range a I a I b c d II III b c d II III IV This diagram represents a function since each element in the domain corresponds to eactl one element in the range. This diagram does not represent a function since one element of the domain corresponds with more than one element of the range; ( ai, ), ( a, III ). Page 44
VERTICAL LINE TEST There is a test called the Vertical Line Test to determine if a relationship given graphicall is a function. If there is even one vertical line that can be drawn which intersects the graph more than once, then there is an -value which is matched with more than one -value, meaning the graph does not represent a function. Eample. Which of the following graphs are graphs of functions? Drawing a vertical line through this graph will onl cross the graph once. It is a function. FUNCTION NOTATION Drawing a vertical line through this graph will cross the graph twice. This is not a function. Drawing a vertical line through this graph will cross the graph at most once. It is a function. Once we know a relationship represents a function, we often change the notation used to emphasize the fact that it is a function. In accordance with the idea of corresponding input and output values, these equations are often represented with the notation f( ) for the output value. This notation is read as f of and signifies the value that corresponds to a given value. Writing = f( ) represents onl a change in how the output is referenced. In Eample 4 above, instead of writing the function as 3, we could have written f ( ) 3. It is important to point out that f( ) does not mean f times. It is a notation that names the function with the first letter (function f ) and then in parentheses, we are given information about what variable is used as the input in the function (variable ). The first letter naming the function can be anthing we want it to be. For eample, ou will often see g ( ), read g of. FINDING DOMAIN AND RANGE We will now identif the domain and range of functions that are represented b graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the -ais. The range is the set of possible output values, which are shown on the -ais.keep in mind that if the graph continues beond the portion of the graph we can see, the domain and range ma be greater than the visible values. Page 45
Note that the domain and range are alwas written from smaller to larger values: from left to right for domain, and from the bottom of the graph to the top of the graph for range. Consider the graph below. We observe that the graph etends horizontall from 5 to the right without bound, so the domain is all real numbers greater than or equal to 5. We write that domain in interval notation as [ 5, ) or in set builder notation as { 5}. The vertical etent of the graph is all range values 5 and below or all real numbers less than or equal to 5. So the range is written in interval notation as (,5] or in set builder notation as { 5}. Eample. Find the domain and range given the following graph. -3 - - - - -3-4 -6 Page 46
Answer: Domain -3 - - - - -3-4 -6 Range Domain: ( 3,] or 3 Range: [ 4,] or 4 Notice that the open circle at ( 3,) indicates that the point ( 3,) does not belong to the graph. The closed circle at (, 4) indicates that the point (, 4) does belong to the graph. Eample 3. Find the domain of the function. 3 Find values of for which the f ( ) 6 denominator is equal to. 6 Solve b factoring ( 3) ( ) Set each factor equal to zero 3 or Solve each equation 3 3 3 or Eclude these values from the domain of All real numbers ecept 3 Our Answer f The notation in the previous eample tells us that can be an value ecept for 3 and. If were one of those two values, the function would be undefined. Eample 4. Find the domain of the function. f ( ) 3 With this function, there are no ecluded values All real numbers or or (, ) Our Solution In the above eample there are no real numbers that make the function undefined. This means an number can be used for. Eample 5. Find the domain of the function. f ( ) 3 Square roots of negative numbers are not real numbers, so we restrict the domain to all real numbers for which the radicand is nonnegative Page 47
3 3 3 3 3 3 Set up an inequalit Solve Our Answer The notation in the above eample states that our variable can be an number greater than or equal to 3. An number smaller than 3 would make the function undefined because the radicand would have a value less than zero. EVALUATING A FUNCTION USING FUNCTION NOTATION Function notation can be used when we want to evaluate a function. A numerical value or an epression is substituted in the place of the input variable in the equation and the output value or epression is determined. This process is shown in the following eamples. Eample 6. Evaluate the function. Let f ( ) 3 4 ; find f ( ). Substitute for in the function f ( ) 3( ) 4( ) Evaluate, using order of operations 3(4) 4( ) Multipl 8 Add f ( ) Our Answer One advantage of using function notation is that instead of asking what is the value of if?, we can now just write Find f ( ). The answer statement f ( ) above tells us the value of the function is when. Eample 7. Evaluate the function. 6 Let h ( ) 3 ; find h (4). Substitute 4 for in the function (4) 6 h(4) 3 Simplif eponent, multipling first 3 3 8 6 Subtract in eponent Evaluate eponent h(4) 9 Our Answer Page 48
Eample 8. Evaluate the function. Let k( a) a 4 ; find k( 7). Substitute 7 for a in the function k( 7) 7 4 Add inside absolute value 3 (3) k( 7) 6 Evaluate absolute value Multipl Our Answer As the above eamples show, the function can take man different forms, but the wa to evaluate the function is alwas the same: replace the variable with what is in parentheses and simplif. We can also substitute epressions into functions using the same process. Often the epressions use the same variable, so it is important to remember each occurrence of the variable is replaced b whatever is in parentheses. Eample 9. Evaluate the function. Let g( ) 4 ; find (3 ) g(3 ) (3 ) g 4 (3 ) 8 4 Eample. Evaluate the function. g. Replace in the function with (3 ) Simplif eponent Our Answer Let p() t t t ; find pt ( ). Replace each t in the function with ( t ) p( t ) ( t ) ( t ) Square binomial t t ( t ) Distribute negative sign t t t Combine like terms p( t ) t t Our Answer It is important to become comfortable with function notation and learn how to use it as we transition into more advanced algebra topics. EVALUATING A FUNCTION FROM ITS GRAPH Finall, in addition to being evaluated algebraicall, functions can be evaluated b identifing input and output values on its graph. To do this, we onl need to find the desired value for the input variable on the -ais, and then moving along a vertical path through that value, note where this path intersects the graph of the function. Upon finding this intersection we now should move along a horizontal path toward the -ais. The value at which this horizontal path intersects the -ais represents the output value for the desired input value. Page 49
Eample. Using the graph of the function f below, find f ( ). - -3-4 -6-3 - - - Locate on the -ais and visualize a vertical path passing through that value. Remember that this is where the input value is equal to. - -3-3 - - - -4-6 Now, notice where this line intersects the graph of the given function. At that point of intersection, visualize a horizontal line passing through and in the direction of the -ais. - -3-4 -3 - - - -6 At this point it should be noted that the horizontal line intersects the -ais at -4. This is the value of the output variable. Therefore, visualizing both the vertical and horizontal lines at the same time identifies the values of both the input and output variables, and in effect finds what we are looking for, f ( ). This is epressl seen in the graph on the net page. Page 5
-3 - - - Therefore, according to the procedure followed, and the vertical and horizontal lines, we see that f ( ) 4. In other words, the function evaluated at is equal to 4. - -3-4 -6 Page 5
Practice Eercises Determine which of the following represent functions. ) 3) ) 4) Specif the domain of each of the following functions. 5) f ( ) 5 6) f ( ) 8 4 7) st () t 8) st () t 9) f ( ) 3 4 ) f ( ) 6 ) f( ) 3 4 ) ( ) 5 Evaluate each function. 3) g 4 ; Find g () ( ) 3 4 9) w( ) 4 ; Find w( 5) 4) g( ) 5 3; Find g () 5) f ( ) 3 ; Find f () ) w( n) 4n 3 ; Find w () ) p( n) 3 n ; Find p ( 7) 6) 7) f 4 ; Find f ( 9) ( ) 9 9 3; Find f () f ( n) n n ) h( n) 4n ; Find hn ( 3) 3) g( ) ; Find g(3 ) 8) f( t) 3 t ; Find f (4) 4) h() t t t ; Find ht ( ) Page 5
Practice Eercises: Section 5. (continued) For each of the functions in problems 5-7, find the following: a) Domain b) Range c) f ( ) d) f () e) the value(s) such that f( ) 5) 6) 7) 5 4 3-4 -3 - - - 3 4 5 - -3-4 9 8 7 6 5 4 3 - -4-3 - - 3 4 - -3-4 5 4 3-4 -3 - - - 3 4 Determine whether each relation is a function. Identif the domain and range for each relation. 8) {(,4),(3,6),(4,6),(9,)} 9) {(, 5),(, 3),(4, ),(6,),(8,3)} 3) {(5,6),(7,),(9, ),(7,)} Page 53
ANSWERS to Practice Eercises ) Yes ) Yes 3) No 4) No 5) (, ) 6),] 7) all real numbers ecept t 8) (, ) 9) (, ) ) [6, ) ) all real numbers ecept and 4 ) all real numbers ecept 5 and 5 3) 4 4) 7 5) 6) 85 7) 7 8) 79 9) 5 ) ) ) 4n 4 3) 3 4) t 3t The Answers to Practice Eercises are continued on the net page. Page 54
ANSWERS to Practice Eercises: Section 5. (continued) 5) a) 3 b) 5 4 c) f ( ) 3 d) f () 4 e) 6) a).5 b) 4 c) f ( ) 3 d) f () e),, 7) a).5 b) 4 c) f ( ) 3 d) f ().5 e) 8) Function; Domain {,3,4,9} Range {,4,6} 9) Function; Domain {,,4,6,8} Range {,4,6} 3) Not a Function; Domain {5,7,9} Range {,,,6} Page 55
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