10. f(x) = 3 2 x f(x) = 3 x 12. f(x) = 1 x 2 + 1

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1 Relations and Functions.6. Eercises To see all of the help resources associated with this section, click OSttS Chapter b. In Eercises -, sketch the graph of the given function. State the domain of the function, identif an intercepts and test for smmetr. For help with these eercises, click on one or more of the resources below: Finding domain Graphing functions b plotting points Determining if a function is even, odd, or neither. f() =. f() =. f() = +. f() =. f() = 6. f() = 7. f() = ( )( + ) 8. f() = 9. f() = 0. f() = +. f() =. f() = + In Eercises - 0, sketch the graph of the given piecewise-defined function. For help with these eercises, click on the resource below: Graphing piecewise defined functions { if. f() = if > if < 0. f() = if 0 < < if > { if < 0 7. f() = if 0 if 9. f() = if < < if {. f() = if 0 if > 0 if 6. f() = if < < if { + if < 8. f() = if if 6 < < 0. f() = if < < if < < 9

2 .6 Graphs of Functions In Eercises -, determine analticall if the following functions are even, odd or neither. For help with these eercises, click on the resource below: Determining if a function is even, odd, or neither. f() = 7. f() = 7 +. f() = 7. f() =. f() = 6. f() = 6 7. f() = 8. f() = + 9. f() = f() = f() =. f() =. f() = 0. f() =. f() = 6. f() = 7. f() = + 9. f() = 0. f() = 8. f() = f() = For help with Eercises - 7, click obtaining information about a function from its graph. In Eercises - 7, use the graph of = f() given below to answer the question.. Find the domain of f.. Find the range of f.. Determine f( ).. Solve f() =.

3 Relations and Functions 6. List the -intercepts, if an eist. 7. List the -intercepts, if an eist. 8. Find the zeros of f. 9. Solve f() Find the number of solutions to f() =.. Does f appear to be even, odd, or neither?. List the intervals where f is increasing.. List the intervals where f is decreasing.. List the local maimums, if an eist.. List the local minimums, if an eist. 6. Find the maimum, if it eists. 7. Find the minimum, if it eists. In Eercises 8-7, use the graph of = f() given below to answer the question. 8. Find the domain of f. 9. Find the range of f. 60. Determine f(). 6. Solve f() =. 6. List the -intercepts, if an eist. 6. List the -intercepts, if an eist. 6. Find the zeros of f. 6. Solve f() Find the number of solutions to f() =. 67. Does f appear to be even, odd, or neither? 68. List the intervals where f is increasing. 69. List the intervals where f is decreasing. 70. List the local maimums, if an eist. 7. List the local minimums, if an eist. 7. Find the maimum, if it eists. 7. Find the minimum, if it eists.

4 .6 Graphs of Functions In Eercises 7-77, use our graphing calculator to approimate the local and absolute etrema of the given function. Approimate the intervals on which the function is increasing and those on which it is decreasing. Round our answers to two decimal places. For help with these eercises, click on resource below: Approimating relative etrema 7. f() = f() = / ( ) 76. f() = f() = 9 In Eercises 78-8, use the graphs of = f() and = g() below to find the function value. = f() = g() 78. (f + g)(0) 79. (f + g)() 80. (f g)() 8. (g f)() ( ) ( ) 8. (fg)() 8. (fg)() 8. f g () 8. g f () The graph below represents the height h of a Sasquatch (in feet) as a function of its age N in ears. Use it to answer the questions in Eercises N = h(n) 86. Find and interpret h(0).

5 6 Relations and Functions 87. How tall is the Sasquatch when she is ears old? 88. Solve h(n) = 6 and interpret. 89. List the interval over which h is constant and interpret our answer. 90. List the interval over which h is decreasing and interpret our answer. For Eercises 9-9, let f() = be the greatest integer function as defined in Eercise 7 in Section.. For help with these eercises, click on the resource below: The greatest integer function 9. Graph = f(). Be careful to correctl describe the behavior of the graph near the integers. 9. Is f even, odd, or neither? Eplain. 9. Discuss with our classmates which points on the graph are local minimums, local maimums or both. Is f ever increasing? Decreasing? Constant? In Eercises 9-9, use our graphing calculator to show that the given function does not have an etrema, neither local nor absolute. 9. f() = + 9. f() = In Eercise 7 in Section., we saw that the population of Sasquatch in Portage Count could be modeled b the function P (t) = 0t, where t = 0 represents the ear 80. Use t + our graphing calculator to analze the general function behavior of P. Will there ever be a time when 00 Sasquatch roam Portage Count? 97. Suppose f and g are both even functions. What can be said about the functions f + g, f g, fg and f g? What if f and g are both odd? What if f is even but g is odd? 98. One of the most important aspects of the Cartesian Coordinate Plane is its abilit to put Algebra into geometric terms and Geometr into algebraic terms. We ve spent most of this chapter looking at this ver phenomenon and now ou should spend some time with our classmates reviewing what we ve done. What major results do we have that tie Algebra and Geometr together? What concepts from Geometr have we not et described algebraicall? What topics from Intermediate Algebra have we not et discussed geometricall?

6 .6 Graphs of Functions 7 It s now time to thoroughl vet the pathologies induced b the precise definitions of local maimum and local minimum. We ll do this b providing ou and our classmates a series of Eercises to discuss. You will need to refer back to Definition.0 (Increasing, Decreasing and Constant) and Definition. (Maimum and Minimum) during the discussion. 99. Consider the graph of the function f given below. (a) Show that f has a local maimum but not a local minimum at the point (, ). (b) Show that f has a local minimum but not a local maimum at the point (, ). (c) Show that f has a local maimum AND a local minimum at the point (0, ). (d) Show that f is constant on the interval [, ] and thus has both a local maimum AND a local minimum at ever point (, f()) where < <. 00. Using Eample.6. as a guide, show that the function g whose graph is given below does not have a local maimum at (, ) nor does it have a local minimum at (, ). Find its etrema, both local and absolute. What s unique about the point (0, ) on this graph? Also find the intervals on which g is increasing and those on which g is decreasing.

7 8 Relations and Functions 0. We said earlier in the section that it is not good enough to sa local etrema eist where a function changes from increasing to decreasing or vice versa. As a previous eercise showed, we could have local etrema when a function is constant so now we need to eamine some functions whose graphs do indeed change direction. Consider the functions graphed below. Notice that all four of them change direction at an open circle on the graph. Eamine each for local etrema. What is the effect of placing the dot on the -ais above or below the open circle? What could ou sa if no function value were assigned to = 0? (a) Function I (b) Function II (c) Function III (d) Function IV Let f() = 0 +. Checkpoint Quiz.6. Find the domain of f. Write our answer using interval notation.. Find the - and -intercepts of the graph of = f().. Is f even, odd, or neither? Eplain.. Use a graphing utilit to graph = f(). From the graph, approimate the intervals over which f is increasing and the intervals over which it is decreasing. List the local maimums and minimums, if the eist, and approimate the range of f. For worked out solutions to this quiz, click the links below: Quiz Solution Part Quiz Solution Part

8 .6 Graphs of Functions 9.6. Answers. f() = -intercept: (, 0) -intercept: (0, ) No smmetr. f() = -intercept: (, 0) -intercept: ( 0, ) No smmetr. f() = + -intercept: None -intercept: (0, ) Even. f() = -intercepts: (, 0), (, 0) -intercept: (0, ) Even. f() = -intercept: None -intercept: (0, ) Even

9 0 Relations and Functions 6. f() = -intercept: (0, 0) -intercept: (0, 0) Odd f() = ( )( + ) -intercepts: (, 0), (0, 0), (, 0) -intercept: (0, 0) No smmetr 8. f() = Domain: [, ) -intercept: (, 0) -intercept: None No smmetr f() = Domain: (, ] -intercept: (, 0) -intercept: (0, ) No smmetr

10 .6 Graphs of Functions 0. f() = + Domain: [, ) -intercept: (, 0) -intercept: (0, ) No smmetr. f() = -intercept: (0, 0) -intercept: (0, 0) Odd f() = + -intercept: None -intercept: (0, ) Even

11 Relations and Functions odd. neither. even. even. even 6. neither 7. odd 8. odd 9. even 0. neither. neither. even. even and odd. odd. even 6. even 7. neither 8. odd 9. odd 0. even. even. [, ]. [, ]. f( ) =. = 6. (, 0), (, 0), (, 0) 7. (0, ) 8.,, 9. [, ], [, ] 0.

12 .6 Graphs of Functions. neither. [, ], [0, ]. [, 0], [, ]. f( ) =, f() =. f(0) = 6. f( ) = 7. f( ) = 8. [, ] 9. [, ) 60. f() = 6. = 6. (, 0), (0, 0), (, 0) 6. (0, 0) 6., 0, 6. [, 0] {} neither 68. [, ) 69. [, ], (, ] 70. none 7. f( ) =, f() = 7. none 7. f( ) = 7. No absolute maimum Absolute minimum f(.) 7.6 Local minimum at (.8, 9.) Local maimum at (0.,.7) Local minimum at (., 7.6) Increasing on [.8, 0.], [., ) Decreasing on (,.8], [0.,.] 76. Absolute maimum f(0) = Absolute minimum f(±) = 0 Local maimum at (0, ) No local minimum Increasing on [, 0] Decreasing on [0, ] 7. No absolute maimum No absolute minimum Local maimum at (0, 0) Local minimum at (.60,.8) Increasing on (, 0], [.60, ) Decreasing on [0,.60] 77. Absolute maimum f(.).0 Absolute minimum f(.).0 Local maimum (.,.0) Local minimum (.,.0) Increasing on [.,.] Decreasing on [,.], [., ] 78. (f + g)(0) = 79. (f + g)() = 80. (f g)() = 8. (g f)() = 0 ( ) ( ) 8. (fg)() = 9 8. (fg)() = 6 8. f g () = 0 8. g f () = 86. h(0) =, so the Sasquatch is feet tall at birth. 87. h() = 6, so the Saquatch is 6 feet tall when she is ears old. 88. h(n) = 6 when N = and N = 60. This means the Sasquatch is 6 feet tall when she is and 60 ears old. 89. h is constant on [0, ]. This means the Sasquatch s height is constant (at 6 feet) for these ears.

13 Relations and Functions 90. h is decreasing on [, 60]. This means the Sasquatch is getting shorter from the age of to the age of 60. (Sasquatchteoporosis, perhaps?) Note that f(.) =, but f(.) =, so f is neither even nor odd The graph of f() =.