 3.2.3.2.1: Find the domain and the range of each relation. Also determine whet...
 3.2.3.2.2: Find the domain and the range of each relation. Also determine whet...
 3.2.3.2.3: Find the domain and the range of each relation. Also determine whet...
 3.2.3.2.4: Find the domain and the range of each relation. Also determine whet...
 3.2.3.2.5: Find the domain and the range of each relation. Also determine whet...
 3.2.3.2.6: Find the domain and the range of each relation. Also determine whet...
 3.2.3.2.7: Find the domain and the range of each relation. Also determine whet...
 3.2.3.2.8: Find the domain and the range of each relation. Also determine whet...
 3.2.3.2.9: Find the domain and the range of each relation. Also determine whet...
 3.2.3.2.10: Find the domain and the range of each relation. Also determine whet...
 3.2.3.2.11: e a 12,14b , a0, 78b , 10.5, p2 f
 3.2.3.2.12: 4251413355 3 4 1 321 2 4 5 xy 17. Inp
 3.2.3.2.13: Input19941998200220062010OutputNumber of Gold Medalswon by U.S.Year...
 3.2.3.2.14: InputPolar BearCowChimpanzeeGiraffeGorillaKangarooRed FoxOutputAnim...
 3.2.3.2.15: Input3210421250Output04010100
 3.2.3.2.16: InputCatDogToOfGivenOutput3541762
 3.2.3.2.17: Input0Output215100
 3.2.3.2.18: Input Output123
 3.2.3.2.19: In Exercises 19 through 22, determine whether the relation is a fun...
 3.2.3.2.20: In Exercises 19 through 22, determine whether the relation is a fun...
 3.2.3.2.21: In Exercises 19 through 22, determine whether the relation is a fun...
 3.2.3.2.22: In Exercises 19 through 22, determine whether the relation is a fun...
 3.2.3.2.23: Use the vertical line test to determine whether each graph is the g...
 3.2.3.2.24: Use the vertical line test to determine whether each graph is the g...
 3.2.3.2.25: Use the vertical line test to determine whether each graph is the g...
 3.2.3.2.26: Use the vertical line test to determine whether each graph is the g...
 3.2.3.2.27: Use the vertical line test to determine whether each graph is the g...
 3.2.3.2.28: Use the vertical line test to determine whether each graph is the g...
 3.2.3.2.29: Find the domain and the range of each relation. Use the vertical li...
 3.2.3.2.30: Find the domain and the range of each relation. Use the vertical li...
 3.2.3.2.31: Find the domain and the range of each relation. Use the vertical li...
 3.2.3.2.32: Find the domain and the range of each relation. Use the vertical li...
 3.2.3.2.33: Find the domain and the range of each relation. Use the vertical li...
 3.2.3.2.34: Find the domain and the range of each relation. Use the vertical li...
 3.2.3.2.35: Find the domain and the range of each relation. Use the vertical li...
 3.2.3.2.36: Find the domain and the range of each relation. Use the vertical li...
 3.2.3.2.37: Find the domain and the range of each relation. Use the vertical li...
 3.2.3.2.38: Find the domain and the range of each relation. Use the vertical li...
 3.2.3.2.39: Find the domain and the range of each relation. Use the vertical li...
 3.2.3.2.40: Find the domain and the range of each relation. Use the vertical li...
 3.2.3.2.41: Decide whether each is a function. See Examples 3 through 6. y = x + 1
 3.2.3.2.42: Decide whether each is a function. See Examples 3 through 6.y = x  1
 3.2.3.2.43: Decide whether each is a function. See Examples 3 through 6.x = 2y2
 3.2.3.2.44: Decide whether each is a function. See Examples 3 through 6. y = x2
 3.2.3.2.45: Decide whether each is a function. See Examples 3 through 6.y  x = 7
 3.2.3.2.46: Decide whether each is a function. See Examples 3 through 6.2x  3y...
 3.2.3.2.47: Decide whether each is a function. See Examples 3 through 6.y = 1x
 3.2.3.2.48: Decide whether each is a function. See Examples 3 through 6.y = 1x  3
 3.2.3.2.49: Decide whether each is a function. See Examples 3 through 6.y = 5x ...
 3.2.3.2.50: Decide whether each is a function. See Examples 3 through 6.y = 12x...
 3.2.3.2.51: Decide whether each is a function. See Examples 3 through 6.x = y2
 3.2.3.2.52: Decide whether each is a function. See Examples 3 through 6.x = 0 y
 3.2.3.2.53: If f 1x2 = 3x + 3, g1x2 = 4x2  6x + 3, and h1x2 = 5x2  7, find th...
 3.2.3.2.54: If f 1x2 = 3x + 3, g1x2 = 4x2  6x + 3, and h1x2 = 5x2  7, find th...
 3.2.3.2.55: If f 1x2 = 3x + 3, g1x2 = 4x2  6x + 3, and h1x2 = 5x2  7, find th...
 3.2.3.2.56: If f 1x2 = 3x + 3, g1x2 = 4x2  6x + 3, and h1x2 = 5x2  7, find th...
 3.2.3.2.57: If f 1x2 = 3x + 3, g1x2 = 4x2  6x + 3, and h1x2 = 5x2  7, find th...
 3.2.3.2.58: If f 1x2 = 3x + 3, g1x2 = 4x2  6x + 3, and h1x2 = 5x2  7, find th...
 3.2.3.2.59: If f 1x2 = 3x + 3, g1x2 = 4x2  6x + 3, and h1x2 = 5x2  7, find th...
 3.2.3.2.60: If f 1x2 = 3x + 3, g1x2 = 4x2  6x + 3, and h1x2 = 5x2  7, find th...
 3.2.3.2.61: Given the following functions, find the indicated values. See Examp...
 3.2.3.2.62: Given the following functions, find the indicated values. See Examp...
 3.2.3.2.63: Given the following functions, find the indicated values. See Examp...
 3.2.3.2.64: Given the following functions, find the indicated values. See Examp...
 3.2.3.2.65: Given the following functions, find the indicated values. See Examp...
 3.2.3.2.66: Given the following functions, find the indicated values. See Examp...
 3.2.3.2.67: Given the following functions, find the indicated values. See Examp...
 3.2.3.2.68: Given the following functions, find the indicated values. See Examp...
 3.2.3.2.69: Use the graph of the functions below to answer Exercises 69 through...
 3.2.3.2.70: Use the graph of the functions below to answer Exercises 69 through...
 3.2.3.2.71: If g142 = 56, write the corresponding ordered pair
 3.2.3.2.72: If g1 22 = 8, write the corresponding ordered pair.
 3.2.3.2.73: Find f 1 12 .
 3.2.3.2.74: Find f 1 22 .
 3.2.3.2.75: Find g(2).
 3.2.3.2.76: Find g1 42 .
 3.2.3.2.77: Find all values of x such that f 1x2 = 5.
 3.2.3.2.78: Find all values of x such that f 1x2 = 2.
 3.2.3.2.79: Find all positive values of x such that g1x2 = 4.
 3.2.3.2.80: Find all values of x such that g1x2 = 0.
 3.2.3.2.81: From the Chapter 3 opener, we have two functions to describe the pe...
 3.2.3.2.82: From the Chapter 3 opener, we have two functions to describe the pe...
 3.2.3.2.83: From the Chapter 3 opener, we have two functions to describe the pe...
 3.2.3.2.84: From the Chapter 3 opener, we have two functions to describe the pe...
 3.2.3.2.85: From the Chapter 3 opener, we have two functions to describe the pe...
 3.2.3.2.86: From the Chapter 3 opener, we have two functions to describe the pe...
 3.2.3.2.87: The function f 1x2 = 0.42x + 10.5, can be used to predict diamond p...
 3.2.3.2.88: The function f 1x2 = 0.42x + 10.5, can be used to predict diamond p...
 3.2.3.2.89: The function f 1x2 = 0.42x + 10.5, can be used to predict diamond p...
 3.2.3.2.90: The function f 1x2 = 0.42x + 10.5, can be used to predict diamond p...
 3.2.3.2.91: The function A1r2 = pr2 may be used to find the area of a circle if...
 3.2.3.2.92: The function A1r2 = pr2 may be used to find the area of a circle if...
 3.2.3.2.93: The function V1x2 = x3 may be used to find the volume of a cube if ...
 3.2.3.2.94: The function V1x2 = x3 may be used to find the volume of a cube if ...
 3.2.3.2.95: Forensic scientists use the following functions to find the height ...
 3.2.3.2.96: Forensic scientists use the following functions to find the height ...
 3.2.3.2.97: The dosage in milligrams D ofIvermectin, a heartworm preventive,for...
 3.2.3.2.98: The dosage in milligrams D ofIvermectin, a heartworm preventive,for...
 3.2.3.2.99: The per capita consumption (in pounds) of all beef in theUnited Sta...
 3.2.3.2.100: The amount of money (in billions of dollars) spent bythe Boeing Com...
 3.2.3.2.101: Complete the given table and use the table to graph the linear equa...
 3.2.3.2.102: Complete the given table and use the table to graph the linear equa...
 3.2.3.2.103: Complete the given table and use the table to graph the linear equa...
 3.2.3.2.104: Complete the given table and use the table to graph the linear equa...
 3.2.3.2.105: Complete the given table and use the table to graph the linear equa...
 3.2.3.2.106: Complete the given table and use the table to graph the linear equa...
 3.2.3.2.107: Is it possible to find the perimeter of the following geometric fig...
 3.2.3.2.108: Is it possible to find the area of the figure in Exercise 107? If s...
 3.2.3.2.109: For Exercises 109 through 112, suppose that y = f 1x2 and it is tru...
 3.2.3.2.110: For Exercises 109 through 112, suppose that y = f 1x2 and it is tru...
 3.2.3.2.111: For Exercises 109 through 112, suppose that y = f 1x2 and it is tru...
 3.2.3.2.112: For Exercises 109 through 112, suppose that y = f 1x2 and it is tru...
 3.2.3.2.113: Given the following functions, find the indicated values.h1x2 = x2 ...
 3.2.3.2.114: Given the following functions, find the indicated values.f 1x2 = x2...
 3.2.3.2.115: Given the following functions, find the indicated values.f 1x2 = 3x...
 3.2.3.2.116: Given the following functions, find the indicated values. f 1x2 = 2...
 3.2.3.2.117: What is the greatest number of xintercepts that a function may hav...
 3.2.3.2.118: What is the greatest number of yintercepts that a function may hav...
 3.2.3.2.119: In your own words, explain how to find the domain of a function giv...
 3.2.3.2.120: Explain the vertical line test and how it is used.
 3.2.3.2.121: Describe a function whose domain is the set of people in your homet...
 3.2.3.2.122: Describe a function whose domain is the set of people in your algeb...
Solutions for Chapter 3.2: Introduction to Functions
Full solutions for Intermediate Algebra  6th Edition
ISBN: 9780321785046
Solutions for Chapter 3.2: Introduction to Functions
Get Full SolutionsSince 122 problems in chapter 3.2: Introduction to Functions have been answered, more than 66690 students have viewed full stepbystep solutions from this chapter. Chapter 3.2: Introduction to Functions includes 122 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·