 46.1: Eric said that if f(x) 5 2 2 x and g(x) 5 x 2 2, then (f 1 g)(x) 5 ...
 46.2: Give an example of a function g for which 2g(x) g(2x). Give an exam...
 46.3: Let f 5 {(0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0)} and g 5 {(...
 46.4: In 47, the graph of y 5 f(x) is shown. a. Find f(0) for each functi...
 46.5: In 47, the graph of y 5 f(x) is shown. a. Find f(0) for each functi...
 46.6: In 47, the graph of y 5 f(x) is shown. a. Find f(0) for each functi...
 46.7: In 47, the graph of y 5 f(x) is shown. a. Find f(0) for each functi...
 46.8: In 813, the domain of f(x) 5 4 2 2x and of g(x) 5 x2 is the set of ...
 46.9: In 813, the domain of f(x) 5 4 2 2x and of g(x) 5 x2 is the set of ...
 46.10: In 813, the domain of f(x) 5 4 2 2x and of g(x) 5 x2 is the set of ...
 46.11: In 813, the domain of f(x) 5 4 2 2x and of g(x) 5 x2 is the set of ...
 46.12: In 813, the domain of f(x) 5 4 2 2x and of g(x) 5 x2 is the set of ...
 46.13: In 813, the domain of f(x) 5 4 2 2x and of g(x) 5 x2 is the set of ...
 46.14: When Brian does odd jobs, he charges ten dollars an hour plus two d...
 46.15: Mrs. Cucci makes candy that she sells for $8.50 a pound. When she s...
 46.16: Create your own function f(x) and show that . Explain why this resu...
Solutions for Chapter 46: The Algebra Of Functions
Full solutions for Amsco's Algebra 2 and Trigonometry  1st Edition
ISBN: 9781567657029
Solutions for Chapter 46: The Algebra Of Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 46: The Algebra Of Functions includes 16 full stepbystep solutions. Since 16 problems in chapter 46: The Algebra Of Functions have been answered, more than 29275 students have viewed full stepbystep solutions from this chapter. Amsco's Algebra 2 and Trigonometry was written by and is associated to the ISBN: 9781567657029. This textbook survival guide was created for the textbook: Amsco's Algebra 2 and Trigonometry, edition: 1.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column space C (A) =
space of all combinations of the columns of A.

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

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

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].

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.