 2.1: In Exercises 1 8, find the slope and intercept (if possible) of th...
 2.2: In Exercises 1 8, find the slope and intercept (if possible) of th...
 2.3: In Exercises 1 8, find the slope and intercept (if possible) of th...
 2.4: In Exercises 1 8, find the slope and intercept (if possible) of th...
 2.5: In Exercises 1 8, find the slope and intercept (if possible) of th...
 2.6: In Exercises 1 8, find the slope and intercept (if possible) of th...
 2.7: In Exercises 1 8, find the slope and intercept (if possible) of th...
 2.8: In Exercises 1 8, find the slope and intercept (if possible) of th...
 2.9: In Exercises 912, plot the points and find the slope of the line pa...
 2.10: In Exercises 912, plot the points and find the slope of the line pa...
 2.11: In Exercises 912, plot the points and find the slope of the line pa...
 2.12: In Exercises 912, plot the points and find the slope of the line pa...
 2.13: In Exercises 1316, find the slopeintercept form of the equation of...
 2.14: In Exercises 1316, find the slopeintercept form of the equation of...
 2.15: In Exercises 1316, find the slopeintercept form of the equation of...
 2.16: In Exercises 1316, find the slopeintercept form of the equation of...
 2.17: In Exercises 1720, find the slopeintercept form of the equation of...
 2.18: In Exercises 1720, find the slopeintercept form of the equation of...
 2.19: In Exercises 1720, find the slopeintercept form of the equation of...
 2.20: In Exercises 1720, find the slopeintercept form of the equation of...
 2.21: In Exercises 21 and 22, write the slopeintercept forms of the equa...
 2.22: In Exercises 21 and 22, write the slopeintercept forms of the equa...
 2.23: RATE OF CHANGE In Exercises 23 and 24, you are given the dollar val...
 2.24: RATE OF CHANGE In Exercises 23 and 24, you are given the dollar val...
 2.25: In Exercises 2528, determine whether the equation represents as a f...
 2.26: In Exercises 2528, determine whether the equation represents as a f...
 2.27: In Exercises 2528, determine whether the equation represents as a f...
 2.28: In Exercises 2528, determine whether the equation represents as a f...
 2.29: In Exercises 2932, evaluate the function at each specified value of...
 2.30: In Exercises 2932, evaluate the function at each specified value of...
 2.31: In Exercises 2932, evaluate the function at each specified value of...
 2.32: In Exercises 2932, evaluate the function at each specified value of...
 2.33: In Exercises 3336, find the domain of the function. Verify your res...
 2.34: In Exercises 3336, find the domain of the function. Verify your res...
 2.35: In Exercises 3336, find the domain of the function. Verify your res...
 2.36: In Exercises 3336, find the domain of the function. Verify your res...
 2.37: PHYSICS The velocity of a ball projected upward from ground level i...
 2.38: MIXTURE PROBLEM From a full 50liter container of a 40% concentrati...
 2.39: In Exercises 39 and 40, find the difference quotient and simplify y...
 2.40: In Exercises 39 and 40, find the difference quotient and simplify y...
 2.41: In Exercises 4144, use the Vertical Line Test to determine whether ...
 2.42: In Exercises 4144, use the Vertical Line Test to determine whether ...
 2.43: In Exercises 4144, use the Vertical Line Test to determine whether ...
 2.44: In Exercises 4144, use the Vertical Line Test to determine whether ...
 2.45: In Exercises 4550, find the zeros of the function algebraically
 2.46: In Exercises 4550, find the zeros of the function algebraically
 2.47: In Exercises 4550, find the zeros of the function algebraically
 2.48: In Exercises 4550, find the zeros of the function algebraically
 2.49: In Exercises 4550, find the zeros of the function algebraically
 2.50: In Exercises 4550, find the zeros of the function algebraically
 2.51: In Exercises 51 and 52, use a graphing utility to graph the functio...
 2.52: In Exercises 51 and 52, use a graphing utility to graph the functio...
 2.53: In Exercises 5356, use a graphing utility to graph the function and...
 2.54: In Exercises 5356, use a graphing utility to graph the function and...
 2.55: In Exercises 5356, use a graphing utility to graph the function and...
 2.56: In Exercises 5356, use a graphing utility to graph the function and...
 2.57: In Exercises 5760, find the average rate of change of the function ...
 2.58: In Exercises 5760, find the average rate of change of the function ...
 2.59: In Exercises 5760, find the average rate of change of the function ...
 2.60: In Exercises 5760, find the average rate of change of the function ...
 2.61: In Exercises 6164, determine whether the function is even, odd, or ...
 2.62: In Exercises 6164, determine whether the function is even, odd, or ...
 2.63: In Exercises 6164, determine whether the function is even, odd, or ...
 2.64: In Exercises 6164, determine whether the function is even, odd, or ...
 2.65: In Exercises 65 and 66, write the linear function such that it has ...
 2.66: In Exercises 65 and 66, write the linear function such that it has ...
 2.67: In Exercises 6778, graph the function.
 2.68: In Exercises 6778, graph the function.
 2.69: In Exercises 6778, graph the function.
 2.70: In Exercises 6778, graph the function.
 2.71: In Exercises 6778, graph the function.
 2.72: In Exercises 6778, graph the function.
 2.73: In Exercises 6778, graph the function.
 2.74: In Exercises 6778, graph the function.
 2.75: In Exercises 6778, graph the function.
 2.76: In Exercises 6778, graph the function.
 2.77: In Exercises 6778, graph the function.
 2.78: In Exercises 6778, graph the function.
 2.79: In Exercises 79 and 80, the figure shows the graph of
 2.80: In Exercises 79 and 80, the figure shows the graph of
 2.81: In Exercises 8194, is related to one of the parent functions descri...
 2.82: In Exercises 8194, is related to one of the parent functions descri...
 2.83: In Exercises 8194, is related to one of the parent functions descri...
 2.84: In Exercises 8194, is related to one of the parent functions descri...
 2.85: In Exercises 8194, is related to one of the parent functions descri...
 2.86: In Exercises 8194, is related to one of the parent functions descri...
 2.87: In Exercises 8194, is related to one of the parent functions descri...
 2.88: In Exercises 8194, is related to one of the parent functions descri...
 2.89: In Exercises 8194, is related to one of the parent functions descri...
 2.90: In Exercises 8194, is related to one of the parent functions descri...
 2.91: In Exercises 8194, is related to one of the parent functions descri...
 2.92: In Exercises 8194, is related to one of the parent functions descri...
 2.93: In Exercises 8194, is related to one of the parent functions descri...
 2.94: In Exercises 8194, is related to one of the parent functions descri...
 2.95: In Exercises 95 and 96, find (a) (b) (c) and (d) What is the domain
 2.96: In Exercises 95 and 96, find (a) (b) (c) and (d) What is the domain
 2.97: In Exercises 97100, find (a) and (b) Find the domain of each functi...
 2.98: In Exercises 97100, find (a) and (b) Find the domain of each functi...
 2.99: In Exercises 97100, find (a) and (b) Find the domain of each functi...
 2.100: In Exercises 97100, find (a) and (b) Find the domain of each functi...
 2.101: In Exercises 101 and 102, find two functions and such that (There a...
 2.102: In Exercises 101 and 102, find two functions and such that (There a...
 2.103: PHONE EXPENDITURES The average annual expenditures (in dollars) for...
 2.104: BACTERIA COUNT The number of bacteria in a refrigerated food is giv...
 2.105: In Exercises 105108, find the inverse function of informally. Verif...
 2.106: In Exercises 105108, find the inverse function of informally. Verif...
 2.107: In Exercises 105108, find the inverse function of informally. Verif...
 2.108: In Exercises 105108, find the inverse function of informally. Verif...
 2.109: In Exercises 109 and 110, determine whether the function has an inv...
 2.110: In Exercises 109 and 110, determine whether the function has an inv...
 2.111: In Exercises 111114, use a graphing utility to graph the function, ...
 2.112: In Exercises 111114, use a graphing utility to graph the function, ...
 2.113: In Exercises 111114, use a graphing utility to graph the function, ...
 2.114: In Exercises 111114, use a graphing utility to graph the function, ...
 2.115: In Exercises 115118, (a) find the inverse function of (b) graph bot...
 2.116: In Exercises 115118, (a) find the inverse function of (b) graph bot...
 2.117: In Exercises 115118, (a) find the inverse function of (b) graph bot...
 2.118: In Exercises 115118, (a) find the inverse function of (b) graph bot...
 2.119: In Exercises 119 and 120, restrict the domain of the function to an...
 2.120: In Exercises 119 and 120, restrict the domain of the function to an...
 2.121: Relative to the graph of the function given by is shifted 9 units t...
 2.122: If and are two inverse functions, then the domain of is equal to th...
 2.123: WRITING Explain how to tell whether a relation between two variable...
 2.124: WRITING Explain the difference between the Vertical Line Test and t...
 2.125: WRITING Describe the basic characteristics of the cubic function. D...
Solutions for Chapter 2: Functions and Their Graphs
Full solutions for College Algebra  8th Edition
ISBN: 9781439048696
Solutions for Chapter 2: Functions and Their Graphs
Get Full SolutionsSince 125 problems in chapter 2: Functions and Their Graphs have been answered, more than 35911 students have viewed full stepbystep solutions from this chapter. Chapter 2: Functions and Their Graphs includes 125 full stepbystep solutions. College Algebra was written by and is associated to the ISBN: 9781439048696. This textbook survival guide was created for the textbook: College Algebra , edition: 8. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.