 2.1: In Exercises 16, determine whether each relation is a function.Give...
 2.2: In Exercises 16, determine whether each relation is a function.Give...
 2.3: In Exercises 16, determine whether each relation is a function.Give...
 2.4: In Exercises 16, determine whether each relation is a function.Give...
 2.5: In Exercises 16, determine whether each relation is a function.Give...
 2.6: In Exercises 16, determine whether each relation is a function.Give...
 2.7: In Exercises 78, determine whether each equation defines y as afunc...
 2.8: In Exercises 78, determine whether each equation defines y as afunc...
 2.9: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.10: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.11: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.12: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.13: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.14: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.15: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.16: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.17: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.18: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.19: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.20: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.21: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.22: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.23: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.24: Use the graph of f to solve Exercises 924. Where applicable, useint...
 2.25: In Exercises 2526, determine whether the graph of each equationis s...
 2.26: In Exercises 2526, determine whether the graph of each equationis s...
 2.27: In Exercises 2738, graph each equation in a rectangular coordinates...
 2.28: In Exercises 2738, graph each equation in a rectangular coordinates...
 2.29: In Exercises 2738, graph each equation in a rectangular coordinates...
 2.30: In Exercises 2738, graph each equation in a rectangular coordinates...
 2.31: In Exercises 2738, graph each equation in a rectangular coordinates...
 2.32: In Exercises 2738, graph each equation in a rectangular coordinates...
 2.33: In Exercises 2738, graph each equation in a rectangular coordinates...
 2.34: In Exercises 2738, graph each equation in a rectangular coordinates...
 2.35: In Exercises 2738, graph each equation in a rectangular coordinates...
 2.36: In Exercises 2738, graph each equation in a rectangular coordinates...
 2.37: In Exercises 2738, graph each equation in a rectangular coordinates...
 2.38: In Exercises 2738, graph each equation in a rectangular coordinates...
 2.39: Let f(x) = 2x2 + x  5.a. Find f(x). Is f even, odd, or neither?b...
 2.40: Let C(x) = b30 if 0 x 20030 + 0.40(x  200) if x 7 200 .a. Find C(1...
 2.41: In Exercises 4144, write a function in slopeintercept form whosegr...
 2.42: In Exercises 4144, write a function in slopeintercept form whosegr...
 2.43: In Exercises 4144, write a function in slopeintercept form whosegr...
 2.44: In Exercises 4144, write a function in slopeintercept form whosegr...
 2.45: Determine whether the line through (2, 4) and (7, 0) isparallel to...
 2.46: Exercise is useful not only in preventing depression, but alsoas a ...
 2.47: Find the average rate of change of f(x) = 3x2  x fromx1 = 1 to x2...
 2.48: In Exercises 4649, give the slope and y@intercept of each linewhose...
 2.49: In Exercises 4649, give the slope and y@intercept of each linewhose...
 2.50: Graph using intercepts: 2x  5y  10 = 0.
 2.51: Graph: 2x  10 = 0.
 2.52: The bar graph shows the average age at which men in theUnited State...
 2.53: The graph shows the percentage of college freshmen whowere liberal ...
 2.54: Find the average rate of change of f(x) = x2  4x fromx1 = 5 to x2 ...
 2.55: In Exercises 5559, use the graph of y = f(x) to graph eachfunction ...
 2.56: In Exercises 5559, use the graph of y = f(x) to graph eachfunction ...
 2.57: In Exercises 5559, use the graph of y = f(x) to graph eachfunction ...
 2.58: In Exercises 5559, use the graph of y = f(x) to graph eachfunction ...
 2.59: In Exercises 5559, use the graph of y = f(x) to graph eachfunction ...
 2.60: In Exercises 6063, begin by graphing the standard quadraticfunction...
 2.61: In Exercises 6063, begin by graphing the standard quadraticfunction...
 2.62: In Exercises 6063, begin by graphing the standard quadraticfunction...
 2.63: In Exercises 6063, begin by graphing the standard quadraticfunction...
 2.64: In Exercises 6466, begin by graphing the square root function,f(x) ...
 2.65: In Exercises 6466, begin by graphing the square root function,f(x) ...
 2.66: In Exercises 6466, begin by graphing the square root function,f(x) ...
 2.67: In Exercises 6769, begin by graphing the absolute value function,f(...
 2.68: In Exercises 6769, begin by graphing the absolute value function,f(...
 2.69: In Exercises 6769, begin by graphing the absolute value function,f(...
 2.70: In Exercises 7072, begin by graphing the standard cubic function,f(...
 2.71: In Exercises 7072, begin by graphing the standard cubic function,f(...
 2.72: In Exercises 7072, begin by graphing the standard cubic function,f(...
 2.73: In Exercises 7375, begin by graphing the cube root function,f(x) = ...
 2.74: In Exercises 7375, begin by graphing the cube root function,f(x) = ...
 2.75: In Exercises 7375, begin by graphing the cube root function,f(x) = ...
 2.76: In Exercises 7681, find the domain of each function.f(x) = x2 + 6x  3
 2.77: In Exercises 7681, find the domain of each function.g(x) = 4x  7
 2.78: In Exercises 7681, find the domain of each function.h(x) = 28  2x
 2.79: In Exercises 7681, find the domain of each function.f(x) = xx2 + 4x...
 2.80: In Exercises 7681, find the domain of each function.g(x) = 2x  2x  5
 2.81: In Exercises 7681, find the domain of each function.f(x) = 2x  1 +...
 2.82: In Exercises 8284, find f + g, f  g, fg, and fg . Determine thedom...
 2.83: In Exercises 8284, find f + g, f  g, fg, and fg . Determine thedom...
 2.84: In Exercises 8284, find f + g, f  g, fg, and fg . Determine thedom...
 2.85: In Exercises 8586, find a. (f g)(x); b. (g f)(x); c. (f g)(3).f(x) ...
 2.86: In Exercises 8586, find a. (f g)(x); b. (g f)(x); c. (f g)(3).f(x) ...
 2.87: In Exercises 8788, find a. (f g)(x); b. the domain of (f g).f(x) = ...
 2.88: In Exercises 8788, find a. (f g)(x); b. the domain of (f g).f(x) = ...
 2.89: In Exercises 8990, express the given function h as a compositionof ...
 2.90: In Exercises 8990, express the given function h as a compositionof ...
 2.91: In Exercises 9192, find f(g(x)) and g(f(x)) and determine whetherea...
 2.92: In Exercises 9192, find f(g(x)) and g(f(x)) and determine whetherea...
 2.93: The functions in Exercises 9395 are all onetoone. For each functi...
 2.94: The functions in Exercises 9395 are all onetoone. For each functi...
 2.95: The functions in Exercises 9395 are all onetoone. For each functi...
 2.96: Which graphs in Exercises 9699 represent functions that haveinverse...
 2.97: Which graphs in Exercises 9699 represent functions that haveinverse...
 2.98: Which graphs in Exercises 9699 represent functions that haveinverse...
 2.99: Which graphs in Exercises 9699 represent functions that haveinverse...
 2.100: Use the graph of f in the figure shown to draw the graph ofits inve...
 2.101: In Exercises 101102, find an equation for f 1(x). Then graph fand ...
 2.102: In Exercises 101102, find an equation for f 1(x). Then graph fand ...
 2.103: In Exercises 103104, find the distance between each pair of points....
 2.104: In Exercises 103104, find the distance between each pair of points....
 2.105: In Exercises 105106, find the midpoint of each line segment withthe...
 2.106: In Exercises 105106, find the midpoint of each line segment withthe...
 2.107: In Exercises 107108, write the standard form of the equation ofthe ...
 2.108: In Exercises 107108, write the standard form of the equation ofthe ...
 2.109: In Exercises 109111, give the center and radius of each circle andg...
 2.110: In Exercises 109111, give the center and radius of each circle andg...
 2.111: In Exercises 109111, give the center and radius of each circle andg...
Solutions for Chapter 2: Functions and Graphs
Full solutions for College Algebra  7th Edition
ISBN: 9780134469164
Solutions for Chapter 2: Functions and Graphs
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra , edition: 7. Since 111 problems in chapter 2: Functions and Graphs have been answered, more than 32433 students have viewed full stepbystep solutions from this chapter. Chapter 2: Functions and Graphs includes 111 full stepbystep solutions. College Algebra was written by and is associated to the ISBN: 9780134469164.

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.

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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·

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.