 3.3.1: The interval can be written as the inequality
 3.3.2: The slope of the line containing the points and is
 3.3.3: Test the equation for symmetry with respect to the xaxis, the yax...
 3.3.4: Write the pointslope form of the line with slope 5 containing the p...
 3.3.5: The intercepts of the equation are
 3.3.6: A function is on an open interval I if, for any choice of and in I,...
 3.3.7: A(n) function is one for which for every x in the domain of a(n) fu...
 3.3.8: True or False A function is decreasing on an open interval I if, fo...
 3.3.9: True or False A function has a local maximum at c if there is an op...
 3.3.10: True or False Even functions have graphs that are symmetric with re...
 3.3.11: In 1120, use the graph of the function given. Is increasing on the ...
 3.3.12: In 1120, use the graph of the function given. Is decreasing on the ...
 3.3.13: In 1120, use the graph of the function given. Is increasing on the ...
 3.3.14: In 1120, use the graph of the function given. Is decreasing on the ...
 3.3.15: In 1120, use the graph of the function given. List the interval(s) ...
 3.3.16: In 1120, use the graph of the function given. List the interval(s) ...
 3.3.17: In 1120, use the graph of the function given. Is there a local maxi...
 3.3.18: In 1120, use the graph of the function given. Is there a local maxi...
 3.3.19: In 1120, use the graph of the function given. List the number(s) at...
 3.3.20: In 1120, use the graph of the function given. List the number(s) at...
 3.3.21: In 2128, the graph of a function is given. Use the graph to find: (...
 3.3.22: In 2128, the graph of a function is given. Use the graph to find: (...
 3.3.23: In 2128, the graph of a function is given. Use the graph to find: (...
 3.3.24: In 2128, the graph of a function is given. Use the graph to find: (...
 3.3.25: In 2128, the graph of a function is given. Use the graph to find: (...
 3.3.26: In 2128, the graph of a function is given. Use the graph to find: (...
 3.3.27: In 2128, the graph of a function is given. Use the graph to find: (...
 3.3.28: In 2128, the graph of a function is given. Use the graph to find: (...
 3.3.29: In 2932, the graph of a function is given. Use the graph to find: (...
 3.3.30: In 2932, the graph of a function is given. Use the graph to find: (...
 3.3.31: In 2932, the graph of a function is given. Use the graph to find: (...
 3.3.32: In 2932, the graph of a function is given. Use the graph to find: (...
 3.3.33: In 3344, determine algebraically whether each function is even, odd...
 3.3.34: In 3344, determine algebraically whether each function is even, odd...
 3.3.35: In 3344, determine algebraically whether each function is even, odd...
 3.3.36: In 3344, determine algebraically whether each function is even, odd...
 3.3.37: In 3344, determine algebraically whether each function is even, odd...
 3.3.38: In 3344, determine algebraically whether each function is even, odd...
 3.3.39: In 3344, determine algebraically whether each function is even, odd...
 3.3.40: In 3344, determine algebraically whether each function is even, odd...
 3.3.41: In 3344, determine algebraically whether each function is even, odd...
 3.3.42: In 3344, determine algebraically whether each function is even, odd...
 3.3.43: In 3344, determine algebraically whether each function is even, odd...
 3.3.44: In 3344, determine algebraically whether each function is even, odd...
 3.3.45: In 4552, for each graph of a function find the absolute maximum and...
 3.3.46: In 4552, for each graph of a function find the absolute maximum and...
 3.3.47: In 4552, for each graph of a function find the absolute maximum and...
 3.3.48: In 4552, for each graph of a function find the absolute maximum and...
 3.3.49: In 4552, for each graph of a function find the absolute maximum and...
 3.3.50: In 4552, for each graph of a function find the absolute maximum and...
 3.3.51: In 4552, for each graph of a function find the absolute maximum and...
 3.3.52: In 4552, for each graph of a function find the absolute maximum and...
 3.3.53: In 5360, use a graphing utility to graph each function over the ind...
 3.3.54: In 5360, use a graphing utility to graph each function over the ind...
 3.3.55: In 5360, use a graphing utility to graph each function over the ind...
 3.3.56: In 5360, use a graphing utility to graph each function over the ind...
 3.3.57: In 5360, use a graphing utility to graph each function over the ind...
 3.3.58: In 5360, use a graphing utility to graph each function over the ind...
 3.3.59: In 5360, use a graphing utility to graph each function over the ind...
 3.3.60: In 5360, use a graphing utility to graph each function over the ind...
 3.3.61: Find the average rate of change of (a) From 0 to 2 (b) From 1 to 3 ...
 3.3.62: Find the average rate of change of (a) From 0 to 2 (b) From 1 to 3 ...
 3.3.63: Find the average rate of change of (a) From to (b) From to 1 (c) Fr...
 3.3.64: Find the average rate of change of (a) From to 1 (b) From 0 to 2 (c...
 3.3.65: (a) Find the average rate of change from 1 to 3. (b) Find an equati...
 3.3.66: (a) Find the average rate of change from 2 to 5. (b) Find an equati...
 3.3.67: (a) Find the average rate of change from to 1. (b) Find an equation...
 3.3.68: (a) Find the average rate of change from to 2. (b) Find an equation...
 3.3.69: (a) Find the average rate of change from 2 to 4. (b) Find an equati...
 3.3.70: (a) Find the average rate of change from 0 to 3. (b) Find an equati...
 3.3.71: (a) Determine whether g is even, odd, or neither. (b) There is a lo...
 3.3.72: (a) Determine whether f is even, odd, or neither. (b) There is a lo...
 3.3.73: (a) Determine whether F is even, odd, or neither. (b) There is a lo...
 3.3.74: (a) Determine whether G is even, odd, or neither. (b) There is a lo...
 3.3.75: Minimum Average Cost The average cost per hour in dollars, , of pro...
 3.3.76: Medicine Concentration The concentration C of a medication in the b...
 3.3.77: Ecoli Growth A strain of Ecoli Beu 397recA441 is placed into a n...
 3.3.78: eFiling Tax Returns The Internal Revenue Service Restructuring and...
 3.3.79: For the function compute each average rate of change: (a) From 0 to...
 3.3.80: For the function compute each average rate of change: (a) From 1 to...
 3.3.81: 8188 require the following discussion of a secant line. The slope o...
 3.3.82: 8188 require the following discussion of a secant line. The slope o...
 3.3.83: 8188 require the following discussion of a secant line. The slope o...
 3.3.84: 8188 require the following discussion of a secant line. The slope o...
 3.3.85: 8188 require the following discussion of a secant line. The slope o...
 3.3.86: 8188 require the following discussion of a secant line. The slope o...
 3.3.87: 8188 require the following discussion of a secant line. The slope o...
 3.3.88: 8188 require the following discussion of a secant line. The slope o...
 3.3.89: Draw the graph of a function that has the following properties: dom...
 3.3.90: Redo with the following additional information: increasing on decre...
 3.3.91: How many xintercepts can a function defined on an interval have if...
 3.3.92: Suppose that a friend of yours does not understand the idea of incr...
 3.3.93: Can a function be both even and odd? Explain.
 3.3.94: Using a graphing utility, graph on the interval Use MAXIMUM to find...
 3.3.95: A function f has a positive average rate of change on the interval ...
 3.3.96: Show that a constant function has an average rate of change of 0. C...
Solutions for Chapter 3.3: Properties of Functions
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter 3.3: Properties of Functions
Get Full SolutionsCollege Algebra was written by and is associated to the ISBN: 9780321716811. This textbook survival guide was created for the textbook: College Algebra, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Since 96 problems in chapter 3.3: Properties of Functions have been answered, more than 32685 students have viewed full stepbystep solutions from this chapter. Chapter 3.3: Properties of Functions includes 96 full stepbystep solutions.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

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.

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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).

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

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

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

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.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.