 3.6.1: How do you solve an absolute value equation?
 3.6.2: How can you tell whether an absolute value function has two xinter...
 3.6.3: When solving an absolute value function, the isolated absolute valu...
 3.6.4: How can you use the graph of an absolute value function to determin...
 3.6.5: Describe all numbers x that are at a distance of 4 from the number ...
 3.6.6: Describe all numbers x that are at a distance of __1 2 from the num...
 3.6.7: Describe the situation in which the distance that point x is from 1...
 3.6.8: Find all function values f(x) such that the distance from f(x) to t...
 3.6.9: For the following exercises, find the x and yintercepts of the gr...
 3.6.10: For the following exercises, find the x and yintercepts of the gr...
 3.6.11: For the following exercises, find the x and yintercepts of the gr...
 3.6.12: For the following exercises, find the x and yintercepts of the gr...
 3.6.13: For the following exercises, find the x and yintercepts of the gr...
 3.6.14: For the following exercises, find the x and yintercepts of the gr...
 3.6.15: For the following exercises, find the x and yintercepts of the gr...
 3.6.16: For the following exercises, graph the absolute value function. Plo...
 3.6.17: For the following exercises, graph the absolute value function. Plo...
 3.6.18: For the following exercises, graph the absolute value function. Plo...
 3.6.19: For the following exercises, graph the given functions by hand. y =...
 3.6.20: For the following exercises, graph the given functions by hand. y =...
 3.6.21: For the following exercises, graph the given functions by hand. y =...
 3.6.22: For the following exercises, graph the given functions by hand. y =...
 3.6.23: For the following exercises, graph the given functions by hand. f(x...
 3.6.24: For the following exercises, graph the given functions by hand. f(x...
 3.6.25: For the following exercises, graph the given functions by hand. f(x...
 3.6.26: For the following exercises, graph the given functions by hand. f(x...
 3.6.27: For the following exercises, graph the given functions by hand. f(x...
 3.6.28: For the following exercises, graph the given functions by hand.f(x)...
 3.6.29: For the following exercises, graph the given functions by hand. f(x...
 3.6.30: For the following exercises, graph the given functions by hand. f(x...
 3.6.31: For the following exercises, graph the given functions by hand. f(x...
 3.6.32: Use a graphing utility to graph f(x) = 10 x 2 on the viewing wind...
 3.6.33: Use a graphing utility to graph f(x) = 100x + 100 on the viewing ...
 3.6.34: For the following exercises, graph each function using a graphing u...
 3.6.35: For the following exercises, graph each function using a graphing u...
 3.6.36: For the following exercises, solve the inequality. If possible, fin...
 3.6.37: For the following exercises, solve the inequality. If possible, fin...
 3.6.38: Cities A and B are on the same eastwest line. Assume that city A i...
 3.6.39: The true proportion p of people who give a favorable rating to Cong...
 3.6.40: Students who score within 18 points of the number 82 will pass a pa...
 3.6.41: A machinist must produce a bearing that is within 0.01 inches of th...
 3.6.42: The tolerance for a ball bearing is 0.01. If the true diameter of t...
Solutions for Chapter 3.6: ABSOLUTE VALUE FUNCTIONS
Full solutions for College Algebra  1st Edition
ISBN: 9781938168383
Solutions for Chapter 3.6: ABSOLUTE VALUE FUNCTIONS
Get Full SolutionsSince 42 problems in chapter 3.6: ABSOLUTE VALUE FUNCTIONS have been answered, more than 30279 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra, edition: 1. College Algebra was written by and is associated to the ISBN: 9781938168383. Chapter 3.6: ABSOLUTE VALUE FUNCTIONS includes 42 full stepbystep solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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!

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Outer product uv T
= column times row = rank one matrix.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.