 4.1: In 15: Determine if the relation is or is not a function. Justify y...
 4.2: In 15: Determine if the relation is or is not a function. Justify y...
 4.3: In 15: Determine if the relation is or is not a function. Justify y...
 4.4: In 15: Determine if the relation is or is not a function. Justify y...
 4.5: In 15: Determine if the relation is or is not a function. Justify y...
 4.6: In 611, each equation defines a function from the set of real numbe...
 4.7: In 611, each equation defines a function from the set of real numbe...
 4.8: In 611, each equation defines a function from the set of real numbe...
 4.9: In 611, each equation defines a function from the set of real numbe...
 4.10: In 611, each equation defines a function from the set of real numbe...
 4.11: In 611, each equation defines a function from the set of real numbe...
 4.12: In 1217, from each graph, determine any rational roots of the funct...
 4.13: In 1217, from each graph, determine any rational roots of the funct...
 4.14: In 1217, from each graph, determine any rational roots of the funct...
 4.15: In 1217, from each graph, determine any rational roots of the funct...
 4.16: In 1217, from each graph, determine any rational roots of the funct...
 4.17: In 1217, from each graph, determine any rational roots of the funct...
 4.18: What are zeros of the function f(x) 5 x 1 3 2 4?
 4.19: In 1922, p 5 {(0, 6), (1, 5), (2, 4), (3, 3), (4, 2)} and q 5 {(0, ...
 4.20: In 1922, p 5 {(0, 6), (1, 5), (2, 4), (3, 3), (4, 2)} and q 5 {(0, ...
 4.21: In 1922, p 5 {(0, 6), (1, 5), (2, 4), (3, 3), (4, 2)} and q 5 {(0, ...
 4.22: In 1922, p 5 {(0, 6), (1, 5), (2, 4), (3, 3), (4, 2)} and q 5 {(0, ...
 4.23: In 2330, for each function f and g, find f(g(x)) and g(f(x)). f(x) ...
 4.24: In 2330, for each function f and g, find f(g(x)) and g(f(x)). f(x) ...
 4.25: In 2330, for each function f and g, find f(g(x)) and g(f(x)). f(x) ...
 4.26: In 2330, for each function f and g, find f(g(x)) and g(f(x)). f(x) ...
 4.27: In 2330, for each function f and g, find f(g(x)) and g(f(x)). f(x) ...
 4.28: In 2330, for each function f and g, find f(g(x)) and g(f(x)). f(x) ...
 4.29: In 2330, for each function f and g, find f(g(x)) and g(f(x)). f(x) ...
 4.30: In 2330, for each function f and g, find f(g(x)) and g(f(x)). f(x) ...
 4.31: In 3136, write the equation of each circle.
 4.32: In 3136, write the equation of each circle.
 4.33: In 3136, write the equation of each circle.
 4.34: In 3136, write the equation of each circle.
 4.35: In 3136, write the equation of each circle.
 4.36: In 3136, write the equation of each circle.
 4.37: The graph of the circle (x 2 2)2 1 (y 1 1)2 5 10 and the line y 5 x...
 4.38: The graph of y 5 x2 is shifted 4 units to the right and 2 units dow...
 4.39: The graph of y 5 x is stretched in the vertical direction by a fa...
 4.40: The graph of y 5 2x 1 3 is reflected in the xaxis. Write an equati...
 4.41: The graph of y 5 x is reflected in the xaxis and then moved 1 un...
 4.42: In 4245: a. Write an equation of the relation formed by interchangi...
 4.43: In 4245: a. Write an equation of the relation formed by interchangi...
 4.44: In 4245: a. Write an equation of the relation formed by interchangi...
 4.45: In 4245: a. Write an equation of the relation formed by interchangi...
Solutions for Chapter 4: Relations And Functions
Full solutions for Amsco's Algebra 2 and Trigonometry  1st Edition
ISBN: 9781567657029
Solutions for Chapter 4: Relations And Functions
Get Full SolutionsAmsco's Algebra 2 and Trigonometry was written by and is associated to the ISBN: 9781567657029. Since 45 problems in chapter 4: Relations And Functions have been answered, more than 28669 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Amsco's Algebra 2 and Trigonometry, edition: 1. Chapter 4: Relations And Functions includes 45 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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.

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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).

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.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of 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.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.