 11.1.1: The graph of a function f is shown below. a. Is f (0) positive or n...
 11.1.2: The graph of a function g is shown below. a. Is g(0) positive or ne...
 11.1.3: Draw the graphs of the power functions p1/3 and p1/4 on the same se...
 11.1.4: Draw the graphs of the power functions p3 and p4 on the same set of...
 11.1.5: Draw the graphs of y = 2x and y = 2x for all real numbers x. What c...
 11.1.6: Graph each of the functions defined in 69 below.
 11.1.7: Graph each of the functions defined in 69 below.
 11.1.8: Graph each of the functions defined in 69 below.
 11.1.9: Graph each of the functions defined in 69 below.
 11.1.10: In each of 1013 a function is defined on a set of integers. Graph e...
 11.1.11: In each of 1013 a function is defined on a set of integers. Graph e...
 11.1.12: In each of 1013 a function is defined on a set of integers. Graph e...
 11.1.13: In each of 1013 a function is defined on a set of integers. Graph e...
 11.1.14: The graph of a function f is shown below. Find the intervals on whi...
 11.1.15: Show that the function f : R R defined by the formula f (x) = 2x 3 ...
 11.1.16: Show that the function g: R R defined by the formula g(x) = (x/3) +...
 11.1.17: Let h be the function from R to R defined by the formula h(x) = x 2...
 11.1.18: Let k: R R be the function defined by the formula k(x) = (x 1)/x fo...
 11.1.19: Show that if a function f : R R is increasing, then f is onetoone.
 11.1.20: Given realvalued functions f and g with the same domain D, the sum...
 11.1.21: a. Let m be any positive integer, and define f (x) = xm for all non...
 11.1.22: Let f be the function whose graph is shown below. Draw the graph of...
 11.1.23: Let h be the function whose graph is shown below. Draw the graph of...
 11.1.24: Let f be a realvalued function of a real variable. Show that if f ...
 11.1.25: Let f be a realvalued function of a real varaible. Show that if f ...
 11.1.26: Let f be a realvalued function of a real variable. Show that if f ...
 11.1.27: In 27 and 28, functions f and g are defined. In each case draw the ...
 11.1.28: In 27 and 28, functions f and g are defined. In each case draw the ...
Solutions for Chapter 11.1: RealValued Functions of a Real Variable and Their Graphs
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 11.1: RealValued Functions of a Real Variable and Their Graphs
Get Full SolutionsDiscrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 11.1: RealValued Functions of a Real Variable and Their Graphs includes 28 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Since 28 problems in chapter 11.1: RealValued Functions of a Real Variable and Their Graphs have been answered, more than 52848 students have viewed full stepbystep solutions from this chapter.

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

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.

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.

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.

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].

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

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

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(D» O.

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

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

Solvable system Ax = b.
The right side b is in the column space of A.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.