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This document contains solutions to several math homework questions about inequalities of combined functions and cost-revenue analysis. Question 1 discusses intervals where two functions are greater than each other based on a graph. Question 2 has students graph two functions and state the same intervals using the graph. Question 3 solves Question 2 algebraically. Question 4 sets up cost and revenue equations for building birdhouses and finds the break-even point. Question 5 modifies the cost equation from Question 4 and discusses the new minimum and maximum number of houses.
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Lecture 3.1 to 3.2 bt
- The document outlines the agenda for today's class which includes announcements, projects due, and an upcoming quiz. The class will cover sections 3.1 and 3.2 on functions.
- It defines what a function is and provides examples. A function assigns each input to exactly one output. It discusses function notation and evaluating functions.
- Additional topics covered include finding domains of functions, piecewise functions, graphing functions using a T-chart, reading graphs, and determining if a relation represents a function using the vertical line test. Worksheets are assigned for classwork and homework.
17, r) -,r I -l19.t... 121.2t-314 23. ^t -rr - .docx
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11 1
Evaluating a Function In Exercises 29-14, evaluate the
function at each specified value of the independent
variable and simplify.
29.fO-3t+t
(a) f(2) (b) /(-4) (c) f(r + 2)
30. s(y) :1 - 3y
(a) s(o) tul s(l) (c) s(s + 2)
tlzt.t(t):t2-2t
(a) h(.2) ft) /,(1.s) (c) h(x + 2)
32. v(r) - !rr3
(a) v(3) (b) Y(;) k) v(2r)
33./(.-r)::-./y
@) f(a) (u) l(0.2s) (c') [email protected])
3a.f(x)- aE+8+2
(a) f (-+) (b) /(8)
1
' x'-9
(a) q(-3) (b) q(z)
)t2+\
36. ./(r)- t'
(il qQ) 0) q(o)
lrl
37. i(r) : "'x
ar f(e) (b) /(-e)
38. -.,. : -r *4
: , -i (b) /(_5)
-, - l. x<0
'lq -, - l. .r > 0'\-
'\-
6r t(0)
', - -:. -r < 0t.
> |
1..
1OG Chapter I Functjons and Their Graphs
Testing for Functions Represented Algebraically In
Exercises 77-28. determine whether the equation
represents,r' as a function of r.
18.x:]2+1
20. y :-lf+ 5
22.r:-.y+5
24.r'l!2:3
20. lyl :1-x
28.r,:8
(c) /(x - 8)
(c) q(y + 3)
(.c) q(.- x)
(c) /(t)
(c) f(t)
(c) JQ)
,cl .f(l)
Evalr.rating a Fr.lnction In Exercises 45-48, assume that
the domain of/is the setA = {-2, - 1, 0, 1, 2}. Determine
the set of ordered pairs representing the function/.
[.rr-4. x<o42.fG)-1r_r,., r>0L1 - i.\
(il f?2) (b) /(0)
[.r- ]. t<0
I
a3.f(x)-1a. 0<r<2
L*, + t. r > 2
(a) .f(.-2) (b) /(1)
(s - ), r < otJ
44. ffrr --]s. us r < I
l.+*- r, r2 l
(a) J(. 2) (b) /(])
(c) /(1)
@ f(a)
(c) f(t)
a6. .f(.x) : x2 - 3
as. /(x) : lx + 1l
45. f(x) - x2
a7. f(x): lxl + z
Evaluating a Function In Exercises 49 and 50, complete
the tahle.
4s. h(t): llr + :l
l" - ?l50..f(r) -:
Finding the lnputs That Have Outputs of Zers In
Exercises 51-54, find all values of x such that/(r) = g'
st. 16) : 15 - 3x 52. f(x): 5r * I
3r-4
sa. f(x') - 2r-3s3. /(x) :
Finding the Dornain of a Function In Exercises 55-6J.
find the domain of the function.
,l ss. fG): 5x2 + 2x - | s6. s(ir) : 7 - 2x2
4-3v
57. hhl - ' 58. ,s( r') -I y-)
se. /(x) - 1C - 1 60. /(x) : X/" + 3x
. t 3 l0
{ el. gtrt - ' - 62. h(r) - .., 1..I f t- i LA
r'*2 -,8+6
64./(:r) :--' o f .t
t -5 -4 -3 -1
It(r)
,t 0 l2
I
2
4
/(')
63. s(.v) : 5- 10
the Domain and Range of a Function In 1)
mriffs 65-68, use a graphing utility to graph the
hhu Find the domain and range of the function.
. ,.-
-E' - \
+ i 66. f(x): 1F I 1
68. g(x) : I, - sl.j1- -r,-; : i1r + 3l
I. Geometry Write the areaA of a circle as a function of
rs --ircumference C.
il" Cmmetry Write the arca A of an equilaterai tiangle
"ts i tunction of the length s of its sides.
1!- E4loration An open box of maximum volume is to
s made from a square piece of mateial, 24 centimeters
cm a side, by cutting equal squares from the corners and
uuia-e up the sides (see figure).
,"1 , The table shows the volume 7 (in cubic centimeters)
of the box for various heights x (in centimeters).
L-se the table to estimate the maximum volume.
i Plot the points (x, I/) from the table in part (a). Does
rtre relation defined.
Banco de preguntas para el ap
This document provides information about Section I, Part A of the Calculus AB exam. It includes 30 multiple choice questions covering topics like limits, derivatives, integrals, and other calculus concepts. A calculator is not allowed for this section. The questions cover skills like evaluating limits, finding derivatives and integrals, solving related rate and optimization problems, and interpreting graphs.
Functions & graphs
1) The document discusses various topics in mathematics including sets, functions, composite functions, exponential and logarithmic graphs, and graph transformations.
2) It provides definitions and examples of sets, functions, and how to represent functions using formulas, arrow diagrams, and graphs. Composite functions are defined as functions of other functions.
3) The document explains how to graph exponential and logarithmic functions and describes the key features of these graphs. It also discusses how different transformations can move a graph in various ways, such as reflecting it or stretching/squashing it.
Day 6 examples
This document provides examples of graphing composite functions. It explains that to find the domain of a composite function f(g(x)), you must consider the restrictions on the domains of both the inner and outer functions. It then gives three examples of determining the composite functions f(g(x)) and g(f(x)) for various functions f and g, and calculating or graphing their domains and ranges. The final example asks to determine possible functions f and g that satisfy three given composite functions.
Day 6 examples
This document provides examples of graphing composite functions. It explains that to find the domain of a composite function f(g(x)), you must consider the restrictions on the domains of both the inner and outer functions. It then gives three examples of determining the composite functions f(g(x)) and g(f(x)) for various functions f and g, and calculating or graphing their domains and ranges. The final example asks to determine possible functions f and g that satisfy three given composite functions.
52 notation and algebra of functions
The document discusses notation and algebra of functions. It defines functions as procedures that assign a unique output to each valid input. Most functions are given by formulas like f(x) = x^2 - 2x + 3, where the input is placed in parentheses and evaluated to get the output. Expressions can be formed by using function notation and evaluating one function's output as the input to another function. An example evaluates two functions f(x) = -3x + 2 and g(x) = -2x^2 - 3x + 1 at different inputs and combines them into one expression.
Functions
This document contains exercises on functions including:
1. Representing the distance a stone falls as a function of time and graphing the relationship.
2. Constructing a linear function model from a table of x and f(x) values.
3. Writing mathematical models for various functions including square, add/multiply, divide/subtract, and ratio functions.
4. Expressing one variable as a function of another from an equation.
5. Determining if sets of ordered pairs form functions.
6. Expressing functions in table, ordered pair, graphical, and mapping forms.
7. Stating if graphs represent functions and writing models for piecewise functions from graphs.
DISTANCE TWO LABELING FOR MULTI-STOREY GRAPHS
An L (2, 1)-labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k }such that |f(x)-f(y)| ≥2 if d(x, y) =1 and | f(x)- f(y)| ≥1 if d(x, y) =2. The L (2, 1)-labeling number λ (G) or span of G is the smallest k such that there is a f with
max {f (v) : vє V(G)}= k. In this paper we introduce a new type of graph called multi-storey graph. The distance two labeling of multi-storey of path, cycle, Star graph, Grid, Planar graph with maximal edges and its span value is determined. Further maximum upper bound span value for Multi-storey of simple
graph are discussed.
3_1 Exponential functions and their graphs (1).ppt
Exponential functions, definituon and examples. Graph and characteristics of exponential functions. Applications of exponential functions.
Similar to Homework (20)
Higher Maths 121 Sets And Functions 1205778086374356 2
Higher Maths 121 Sets And Functions 1205778086374356 2
17, r) -,r I -l19.t... 121.2t-314 23. ^t -rr - .docx
17, r) -,r I -l19.t... 121.2t-314 23. ^t -rr - .docx
3_1 Exponential functions and their graphs (1).ppt
3_1 Exponential functions and their graphs (1).ppt
Homework
- 1. INEQUALITIES OF COMBINED FUNCTIONS HOMEWORK
- 2. QUESTION # 1 Observe the following graph. State the intervals where: F(x)>g(x) and g(x)>f(x)
- 3. SOLUTION # 1 f(x)>g(x) when x<0, 0<x<1 and g(x)>f(x) when x>1
- 4. QUESTION # 2 Using Graphing Technology (Ti-83, GraphCalc, Online Graphing calculator.) Graph the function f(x) and g(x) on the same set of axes. State where f(x)>g(x) and where g(x)>f(x) by looking at the graph. 𝑓𝑥=𝑥−2, 𝑔𝑥=(𝑥−2)4
- 5. SOLUTION # 2 F(x)>g(x) on the interval (2,3) and g(x)>f(x) (-∞,2) and (3,∞)
- 6. QUESTION # 3 Solve question # 2 using a different method learned in class. Only state where f(x)>g(x)
- 7. SOLUTION # 3 𝑓𝑥>𝑔(𝑥) 𝑓𝑥−𝑔𝑥>0 𝑥−2𝑥4−8𝑥3+24𝑥62−32𝑥+16>0 𝑥−2−𝑥4+8𝑥3−24𝑥2+32𝑥−16>0 −𝑥4+8𝑥3−24𝑥2+33𝑥−18>0 Synthetic division (𝑥−3)(−𝑥3+5𝑥−9𝑥+6) (𝑥−2)(𝑥−3)(−𝑥2+3𝑥−3) ∴ f(x)>g(x) on the interval (2,3) Cant be factored 𝑥=2, 𝑥=3
- 8. QUESTION # 4 Claire builds and sells birdhouses. Claire makes n birdhouses in a given week and sells them for 45-n dollars per birdhouse. Her weekly costs include a fixed cost of $280 plus $8 per birdhouse made. Assume that Claire sells all of the birdhouses that she makes. A) Write an equation to represent her total weekly cost. B) Write an equation to represent her total weekly revenue. C) Write an inequality to express the condition for which Claire will make a profit. D) How many birdhouses should Claire build each week in order to make a profit.
- 9. SOLUTION # 4 C(n)=(280+8n) R(n)=(45-n)n R(n)>C(n) 45−𝑛𝑛>280+8 45𝑛−𝑛2−8𝑛−280>0 −𝑛2+37𝑛−280>0 𝑛=37±372−4(−1)(−280)2(1) 𝑛=37±2492 𝑛=26, 𝑥=11 ∴11≤n≤26
- 10. QUESTION # 5 Suppose that the developer (refer to page 450 for full question), found a way to reduce their variable cost to $58 000 per house What would the new cost equation be? How would this effect the minimum and maximum number of houses the developer could build.
- 11. SOLUTION # 6 C(n)=0.058x+8 It increase the maximum and decreases the minimum