 2.1: Which of the following sets of ordered pairs are functions? Give re...
 2.2: Use the vertical line test to determine which of the following rela...
 2.3: Will the graph of a straight line always be a function? Give eviden...
 2.4: Give algebraic evidence to show that the relation x2 + y2 = 9 is no...
 2.5: If F(x)=2x2 + 3x 1, find in simplest form: a F(x + 4) b F(2 x) c F(...
 2.6: Suppose G(x) = 2x + 3 x 4 . a Evaluate: i G(2) ii G(0) iii G(1 2 ) ...
 2.7: f represents a function. What is the difference in meaning between ...
 2.8: The value of a photocopier t years after purchase is given by V (t)...
 2.9: On the same set of axes draw the graphs of three different function...
 2.10: Find a linear function f(x) = ax + b for which f(2) = 1 and f(3) = 11
 2.11: Given f(x) = ax + b x , f(1) = 1, and f(2) = 5, find constants a an...
 2.12: Given T(x) = ax2 + bx + c, T(0) = 4, T(1) = 2, and T(2) = 6, find a...
 2.13: Consider the functions f : x 7! 5x and g : x 7! px . a Find: i f(2)...
 2.14: Given f : x 7! 2x and g : x 7! 4x 3, show that (f1 g1)(x)=(g f)1(x).
 2.15: Which of these functions is a selfinverse function? a f(x)=2x b f(...
 2.16: The horizontal line test says: For a function to have an inverse fu...
Solutions for Chapter 2: RELATIONS AND FUNCTIONS
Full solutions for Mathematics for the International Student: Mathematics SL  3rd Edition
ISBN: 9781921972089
Solutions for Chapter 2: RELATIONS AND FUNCTIONS
Get Full SolutionsMathematics for the International Student: Mathematics SL was written by and is associated to the ISBN: 9781921972089. Chapter 2: RELATIONS AND FUNCTIONS includes 16 full stepbystep solutions. Since 16 problems in chapter 2: RELATIONS AND FUNCTIONS have been answered, more than 11164 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: Mathematics for the International Student: Mathematics SL, edition: 3.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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.)

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

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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).

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).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.