 1.3.1.3.1: The graph of a function is a collection of _______ such that is in ...
 1.3.1.3.2: The _______ is used to determine whether the graph of an equation i...
 1.3.1.3.3: A function is _______ on an interval if, for any and in the interva...
 1.3.1.3.4: A function value is a relative _______ of if there exists an interv...
 1.3.1.3.5: The function is called the _______ function, and is an example of a...
 1.3.1.3.6: A function is _______ if, for each in the domain of f x f, fx fx. fx
 1.3.1.3.7: In Exercises 510, use a graphing utility to graph the function and ...
 1.3.1.3.8: In Exercises 510, use a graphing utility to graph the function and ...
 1.3.1.3.9: In Exercises 510, use a graphing utility to graph the function and ...
 1.3.1.3.10: In Exercises 510, use a graphing utility to graph the function and ...
 1.3.1.3.11: In Exercises 1114, use the given function to answer the questions. ...
 1.3.1.3.12: In Exercises 1114, use the given function to answer the questions. ...
 1.3.1.3.13: In Exercises 1114, use the given function to answer the questions. ...
 1.3.1.3.14: In Exercises 1114, use the given function to answer the questions. ...
 1.3.1.3.15: In Exercises 1518, use the Vertical Line Test to determine whether ...
 1.3.1.3.16: In Exercises 1518, use the Vertical Line Test to determine whether ...
 1.3.1.3.17: In Exercises 1518, use the Vertical Line Test to determine whether ...
 1.3.1.3.18: In Exercises 1518, use the Vertical Line Test to determine whether ...
 1.3.1.3.19: In Exercises 1922, determine the open intervals over which the func...
 1.3.1.3.20: In Exercises 1922, determine the open intervals over which the func...
 1.3.1.3.21: In Exercises 1922, determine the open intervals over which the func...
 1.3.1.3.22: In Exercises 1922, determine the open intervals over which the func...
 1.3.1.3.23: In Exercises 2330, (a) use a graphing utility to graph the function...
 1.3.1.3.24: In Exercises 2330, (a) use a graphing utility to graph the function...
 1.3.1.3.25: In Exercises 2330, (a) use a graphing utility to graph the function...
 1.3.1.3.26: In Exercises 2330, (a) use a graphing utility to graph the function...
 1.3.1.3.27: In Exercises 2330, (a) use a graphing utility to graph the function...
 1.3.1.3.28: In Exercises 2330, (a) use a graphing utility to graph the function...
 1.3.1.3.29: In Exercises 2330, (a) use a graphing utility to graph the function...
 1.3.1.3.30: In Exercises 2330, (a) use a graphing utility to graph the function...
 1.3.1.3.31: In Exercises 3136, use a graphing utility to approximate any relati...
 1.3.1.3.32: In Exercises 3136, use a graphing utility to approximate any relati...
 1.3.1.3.33: In Exercises 3136, use a graphing utility to approximate any relati...
 1.3.1.3.34: In Exercises 3136, use a graphing utility to approximate any relati...
 1.3.1.3.35: In Exercises 3136, use a graphing utility to approximate any relati...
 1.3.1.3.36: In Exercises 3136, use a graphing utility to approximate any relati...
 1.3.1.3.37: In Exercises 3742, (a) approximate the relative minimum or relative...
 1.3.1.3.38: In Exercises 3742, (a) approximate the relative minimum or relative...
 1.3.1.3.39: In Exercises 3742, (a) approximate the relative minimum or relative...
 1.3.1.3.40: In Exercises 3742, (a) approximate the relative minimum or relative...
 1.3.1.3.41: In Exercises 3742, (a) approximate the relative minimum or relative...
 1.3.1.3.42: In Exercises 3742, (a) approximate the relative minimum or relative...
 1.3.1.3.43: In Exercises 4350, sketch the graph of the piecewisedefined functio...
 1.3.1.3.44: In Exercises 4350, sketch the graph of the piecewisedefined functio...
 1.3.1.3.45: In Exercises 4350, sketch the graph of the piecewisedefined functio...
 1.3.1.3.46: In Exercises 4350, sketch the graph of the piecewisedefined functio...
 1.3.1.3.47: In Exercises 4350, sketch the graph of the piecewisedefined functio...
 1.3.1.3.48: In Exercises 4350, sketch the graph of the piecewisedefined functio...
 1.3.1.3.49: In Exercises 4350, sketch the graph of the piecewisedefined functio...
 1.3.1.3.50: In Exercises 4350, sketch the graph of the piecewisedefined functio...
 1.3.1.3.51: Library of Parent Functions In Exercises 5156, sketch the graph of ...
 1.3.1.3.52: Library of Parent Functions In Exercises 5156, sketch the graph of ...
 1.3.1.3.53: Library of Parent Functions In Exercises 5156, sketch the graph of ...
 1.3.1.3.54: Library of Parent Functions In Exercises 5156, sketch the graph of ...
 1.3.1.3.55: Library of Parent Functions In Exercises 5156, sketch the graph of ...
 1.3.1.3.56: Library of Parent Functions In Exercises 5156, sketch the graph of ...
 1.3.1.3.57: In Exercises 57 and 58, use a graphing utility to graph the functio...
 1.3.1.3.58: In Exercises 57 and 58, use a graphing utility to graph the functio...
 1.3.1.3.59: In Exercises 5966, algebraically determine whether the function is ...
 1.3.1.3.60: In Exercises 5966, algebraically determine whether the function is ...
 1.3.1.3.61: In Exercises 5966, algebraically determine whether the function is ...
 1.3.1.3.62: In Exercises 5966, algebraically determine whether the function is ...
 1.3.1.3.63: In Exercises 5966, algebraically determine whether the function is ...
 1.3.1.3.64: In Exercises 5966, algebraically determine whether the function is ...
 1.3.1.3.65: In Exercises 5966, algebraically determine whether the function is ...
 1.3.1.3.66: In Exercises 5966, algebraically determine whether the function is ...
 1.3.1.3.67: Think About It In Exercises 6772, find the coordinates of a second ...
 1.3.1.3.68: Think About It In Exercises 6772, find the coordinates of a second ...
 1.3.1.3.69: Think About It In Exercises 6772, find the coordinates of a second ...
 1.3.1.3.70: Think About It In Exercises 6772, find the coordinates of a second ...
 1.3.1.3.71: Think About It In Exercises 6772, find the coordinates of a second ...
 1.3.1.3.72: Think About It In Exercises 6772, find the coordinates of a second ...
 1.3.1.3.73: In Exercises 7382, use a graphing utility to graph the function and...
 1.3.1.3.74: In Exercises 7382, use a graphing utility to graph the function and...
 1.3.1.3.75: In Exercises 7382, use a graphing utility to graph the function and...
 1.3.1.3.76: In Exercises 7382, use a graphing utility to graph the function and...
 1.3.1.3.77: In Exercises 7382, use a graphing utility to graph the function and...
 1.3.1.3.78: In Exercises 7382, use a graphing utility to graph the function and...
 1.3.1.3.79: In Exercises 7382, use a graphing utility to graph the function and...
 1.3.1.3.80: In Exercises 7382, use a graphing utility to graph the function and...
 1.3.1.3.81: In Exercises 7382, use a graphing utility to graph the function and...
 1.3.1.3.82: In Exercises 7382, use a graphing utility to graph the function and...
 1.3.1.3.83: In Exercises 8386, graph the function and determine the interval(s)...
 1.3.1.3.84: In Exercises 8386, graph the function and determine the interval(s)...
 1.3.1.3.85: In Exercises 8386, graph the function and determine the interval(s)...
 1.3.1.3.86: In Exercises 8386, graph the function and determine the interval(s)...
 1.3.1.3.87: Communications The cost of using a telephone calling card is $1.05 ...
 1.3.1.3.88: Delivery Charges The cost of sending an overnight package from New ...
 1.3.1.3.89: In Exercises 89 and 90, write the height h of the rectangle as a fu...
 1.3.1.3.90: In Exercises 89 and 90, write the height h of the rectangle as a fu...
 1.3.1.3.91: Population During a 14 year period from 1990 to 2004, the populatio...
 1.3.1.3.92: Fluid Flow The intake pipe of a 100gallon tank has a flow rate of ...
 1.3.1.3.93: True or False? In Exercises 93 and 94, determine whether the statem...
 1.3.1.3.94: True or False? In Exercises 93 and 94, determine whether the statem...
 1.3.1.3.95: Think About It In Exercises 95100, match the graph of the function ...
 1.3.1.3.96: Think About It In Exercises 95100, match the graph of the function ...
 1.3.1.3.97: Think About It In Exercises 95100, match the graph of the function ...
 1.3.1.3.98: Think About It In Exercises 95100, match the graph of the function ...
 1.3.1.3.99: Think About It In Exercises 95100, match the graph of the function ...
 1.3.1.3.100: Think About It In Exercises 95100, match the graph of the function ...
 1.3.1.3.101: Proof Prove that a function of the following form is odd.y a2n1x 2n...
 1.3.1.3.102: Proof Prove that a function of the following form is even.y a2n x 2...
 1.3.1.3.103: If is an even function, determine if is even, odd, or neither. Expl...
 1.3.1.3.104: Think About It Does the graph in Exercise 16 represent as a functio...
 1.3.1.3.105: Think About It Does the graph in Exercise 17 represent as a functio...
 1.3.1.3.106: Writing Write a short paragraph describing three different function...
 1.3.1.3.107: In Exercises 107110, identify the terms. Then identify the coeffici...
 1.3.1.3.108: In Exercises 107110, identify the terms. Then identify the coeffici...
 1.3.1.3.109: In Exercises 107110, identify the terms. Then identify the coeffici...
 1.3.1.3.110: In Exercises 107110, identify the terms. Then identify the coeffici...
 1.3.1.3.111: In Exercises 111114, find (a) the distance between the two points a...
 1.3.1.3.112: In Exercises 111114, find (a) the distance between the two points a...
 1.3.1.3.113: In Exercises 111114, find (a) the distance between the two points a...
 1.3.1.3.114: In Exercises 111114, find (a) the distance between the two points a...
 1.3.1.3.115: In Exercises 115118, evaluate the function at each specified value ...
 1.3.1.3.116: In Exercises 115118, evaluate the function at each specified value ...
 1.3.1.3.117: In Exercises 115118, evaluate the function at each specified value ...
 1.3.1.3.118: In Exercises 115118, evaluate the function at each specified value ...
 1.3.1.3.119: In Exercises 119 and 120, find the difference quotient and simplify...
 1.3.1.3.120: In Exercises 119 and 120, find the difference quotient and simplify...
Solutions for Chapter 1.3: Graphs of Functions
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 1.3: Graphs of Functions
Get Full SolutionsChapter 1.3: Graphs of Functions includes 120 full stepbystep solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Since 120 problems in chapter 1.3: Graphs of Functions have been answered, more than 36202 students have viewed full stepbystep solutions from this chapter.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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].

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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).

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

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

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

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).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.