 42.1: Let f(x) 5 x2 and g(x 1 2) 5 (x 1 2)2 . Are f and g the same functi...
 42.2: Let f(x) 5 x2 and g(x 1 2) 5 x2 1 2. Are f and g the same function?...
 42.3: In 38, for each function: a. Write an expression for f(x). b. Find ...
 42.4: In 38, for each function: a. Write an expression for f(x). b. Find ...
 42.5: In 38, for each function: a. Write an expression for f(x). b. Find ...
 42.6: In 38, for each function: a. Write an expression for f(x). b. Find ...
 42.7: In 38, for each function: a. Write an expression for f(x). b. Find ...
 42.8: In 38, for each function: a. Write an expression for f(x). b. Find ...
 42.9: In 914, y 5 f(x). Find f(23) for each function. f(x) 5 7 2 x
 42.10: In 914, y 5 f(x). Find f(23) for each function. y 5 1 1 x2
 42.11: In 914, y 5 f(x). Find f(23) for each function. f(x) 5 24x
 42.12: In 914, y 5 f(x). Find f(23) for each function. f(x) 5 4
 42.13: In 914, y 5 f(x). Find f(23) for each function. y 5
 42.14: In 914, y 5 f(x). Find f(23) for each function. f(x) 5 x 1 2
 42.15: The graph of the function f is shown at the right. Find: a. f(2) b....
 42.16: The sales tax t on a purchase is a function of the amount a of the ...
 42.17: A muffin shops weekly profit is a function of the number of muffins...
Solutions for Chapter 42: Function Notation
Full solutions for Amsco's Algebra 2 and Trigonometry  1st Edition
ISBN: 9781567657029
Solutions for Chapter 42: Function Notation
Get Full SolutionsAmsco's Algebra 2 and Trigonometry was written by and is associated to the ISBN: 9781567657029. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 42: Function Notation includes 17 full stepbystep solutions. This textbook survival guide was created for the textbook: Amsco's Algebra 2 and Trigonometry, edition: 1. Since 17 problems in chapter 42: Function Notation have been answered, more than 30730 students have viewed full stepbystep solutions from this chapter.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.

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.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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).

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).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

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).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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 ().

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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