Rational Functions with Removable Discontinuities 1. a) Simplif the rational epression and state an values of where the epression is b) Using the simplified epression in part (a), predict the shape for the graph of the function f( ). Sketch our prediction. c) Graph the function f( ) on a standard ( 10 10) and ( 10 10) window using our graphing calculator. (Enter the function on our calculator as it is written-not in simplified form.) Does the calculator graph appear the same as our graph in part b? d) Turn off the aes on our graphing calculator and graph the function again. What do ou observe? Wh does this happen? e) Draw an open circle on our graph in part b where the graph appears to have a hole.. a) Simplif the rational epression b) Sketch a graph for the function. and state an values of where the epression is Do ou think that the graph of the function will have a hole in it? Draw an open circle on the graph where ou think the hole will occur. Turn off the aes on ou graphing calculator and graph the function using a standard window. Compare the graph on the calculator to our graph in part b. c) Use the trace or the table feature on our calculator to help ou find the value of where the function is discontinuous. Eplain wh ou think that the function is discontinuous at this value of. 46
d) Construct a table of values that shows values of f() when is close to zero. Use a graphing calculator to help ou construct the table of values. < 0 f() > 0 f() -.5.5 -.1.1 -.01.01 -.001.001 -.0005.0005 -.0001.0001 e) As gets closer to 0, what value does f() appear to be approaching?. a) Simplif the rational epression b) Sketch a graph for the function and state an values of where the epression is f( ). Do ou think that the graph of the function will have a hole in it? Draw an open circle on the graph where ou think the hole will occur. Turn off the aes on our graphing calculator and graph the function using a standard window. Compare the graph on the calculator to ou graph in part b. c) Construct a table of values that shows values of f() when is close to zero. Use a graphing calculator to help ou construct the table of values. < 0 f() > 0 f() -.5.5 -.1.1 -.01.01 -.001.001 47
d) As gets closer to 0, what value does f() appear to be approaching? e) Find the domain and range of f(). f) State the intervals of where f() is continuous. 4. a) Simplif the rational epression + 4+ and state an values of where the epression is + + 4+ b) Sketch a graph for the function. Show an points where the graph is + discontinuous with an open circle. c) Verif our graph in part b using a graphing calculator. d) Construct a table of values that shows values of f() when is close to -. Use a graphing calculator to help ou construct the table of values, or the epression for f() ou found in a). < - f() > - f() -.1 -.9 -.01 -.99 -.001 -.999 -.0001 -.9999 -.00001 -.99999 e) As gets closer to -, what value does f() appear to be approaching? f) Find the domain and the range of f(). g) State the intervals of where the f() is continuous. h) This tpe of discontinuit is called removable because we could remove the discontinuit b redefining the function at just one number. 48
For eample: The function + 4+ is discontinuous at -. + WINDOW min -4 ma scl 1 min -4 ma scl 1 B redefining f() at the number -, g() is a continuous function. The function + + 4 + if g ( ) if is continuous. Note: g() + 1 for all. WINDOW min -4 ma scl 1 min -4 ma scl 1 49
5. Match each graph to the correct function. f ( ) g ( ) 4 + h ( ) 4 + a) b) c) For problems 6-8, complete the questions listed below. a) Factor the numerator and denominator of the function and state the domain. b) Reduce an common factors and state an value(s) of where the function is c) State the range of the function. d) Graph the function using paper and pencil. Draw an open circle and label an removable discontinuit on the graph. e) Use our graphing calculator to check our answers. 6. 7. f( ) f( ) 10 + 9 4 + 4 4 + 1 8. f( ) + 6 19 10 50