 21.1: D R 3 2 6 1 5
 21.2: x y 5 2 10 2 15 2 20 2
 21.3: y O x (3, 1) (1, 4) (2, 2) (2, 3)
 21.4: Identify the domain and range. Assume that the January temperatures...
 21.5: Write a relation of ordered pairs for the data.
 21.6: Graph the relation. Is this relation a function?
 21.7: {(7, 8), (7, 5), (7, 2), (7, 1)}
 21.8: {(6, 2.5), (3, 2.5), (4, 2.5)}
 21.9: y = 2x + 1
 21.10: x = y2
 21.11: Find f(5) if f(x) = x2  3x.
 21.12: Find h(2) if h(x) = x 4 3 + 1.
 21.13: D R 10 20 30 1 2 3
 21.14: D R 3 2 1 1 3 5 7
 21.15: x y 0.5 3 2 0.8 0.5 8
 21.16: x y 2000 $4000 2001 $4300 2002 $4600 2003 $4500
 21.17: y x
 21.18: y x
 21.19: O x f(
 21.20: f(x)
 21.21: {(3, 0), (1, 1), (1, 3)} 2
 21.22: {(3, 0), (1, 1), (1, 3)} 2
 21.23: {(2, 1), (3, 0), (1, 5)}
 21.24: {(4, 5), (6, 5), (3, 5)}
 21.25: {(2, 5), (3, 7), (2, 8)}
 21.26: {(3, 4), (4, 3), (6, 5), (5, 6)}
 21.27: {(0, 1.1), (2, 3), (1.4, 2), (3.6, 8)}
 21.28: {(2.5, 1), (1, 1), (0, 1), (1, 1)}
 21.29: y = 5x
 21.30: y = 3x
 21.31: y = 3x  4
 21.32: y = 7x  6
 21.33: y = x2
 21.34: x = 2y2  3
 21.35: f(3)
 21.36: g(3)
 21.37: g (_1 3)
 21.38: f (_2 3)
 21.39: f(a)
 21.40: g(5n)
 21.41: Find the value of f(x) = 3x + 2 when x = 2.
 21.42: What is g(4) if g(x) = x2  5?
 21.43: Make a graph of the data with home runs on the horizontal axis and ...
 21.44: Identify the domain and range.
 21.45: Does the graph represent a function? Explain your reasoning.
 21.46: Write a relation to represent the data.
 21.47: Graph the relation.
 21.48: Identify the domain and range.
 21.49: Is the relation a function? Explain your reasoning
 21.50: Write a relation to represent the data.
 21.51: Graph the relation.
 21.52: Identify the domain and range. Determine whether the relation is di...
 21.53: Is the relation a function? Explain your reasoning.
 21.54: AUDIO BOOK DOWNLOADS Chaz has a collection of 15 audio books. After...
 21.55: OPEN ENDED Write a relation of four ordered pairs that is not a fun...
 21.56: FIND THE ERROR Teisha and Molly are finding g(2a) for the function ...
 21.57: CHALLENGE If f(3a  1) = 12a  7, find one possible expression for ...
 21.58: Writing in Math Use the information about animal lifetimes on page ...
 21.59: ACT/SAT If g(x) = x2, which expression is equal to g(x + 1)? A 1 B ...
 21.60: REVIEW Which set of dimensions represent a triangle similar to the ...
 21.61: y + 1 < 7
 21.62: 5  m < 1
 21.63: x  5 < 0.1
 21.64: SHOPPING Javier had $25.04 when he went to the mall. His friend Sal...
 21.65: 32(22  12) + 42
 21.66: 3(5a + 6b) + 8(2a  b)
 21.67: x + 3 = 2
 21.68: 4 + 2y = 0
 21.69: 0 = _1 2 x  3
 21.70: 1 3 x  4 = 1
Solutions for Chapter 21: Relations and Functions
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 21: Relations and Functions
Get Full SolutionsChapter 21: Relations and Functions includes 70 full stepbystep solutions. Since 70 problems in chapter 21: Relations and Functions have been answered, more than 56336 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302.

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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

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].

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

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

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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

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

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

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

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).

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.