 4.5.1: Determine the domain of the function.f 1x2 = x22  x
 4.5.2: Determine the domain of the function.f 1x2 = 1x3
 4.5.3: Determine the domain of the function.f 1x2 = x + 1x2  6x + 5
 4.5.4: Determine the domain of the function. 1x2 = 1x + 4224x  3
 4.5.5: Determine the domain of the function.f 1x2 = 3x  43x + 15
 4.5.6: Determine the domain of the function.f 1x2 = x2 + 3x  10x2 + 2x
 4.5.7: In Exercises 712, use your knowledge of asymptotes and intercepts t...
 4.5.8: In Exercises 712, use your knowledge of asymptotes and intercepts t...
 4.5.9: In Exercises 712, use your knowledge of asymptotes and intercepts t...
 4.5.10: In Exercises 712, use your knowledge of asymptotes and intercepts t...
 4.5.11: In Exercises 712, use your knowledge of asymptotes and intercepts t...
 4.5.12: In Exercises 712, use your knowledge of asymptotes and intercepts t...
 4.5.13: Determine the vertical asymptotes of the graph of the function.g1x2...
 4.5.14: Determine the vertical asymptotes of the graph of the function.f 1x...
 4.5.15: Determine the vertical asymptotes of the graph of the function.h1x2...
 4.5.16: Determine the vertical asymptotes of the graph of the function. g1x...
 4.5.17: Determine the vertical asymptotes of the graph of the function.f 1x...
 4.5.18: Determine the vertical asymptotes of the graph of the function.h1x2...
 4.5.19: Determine the vertical asymptotes of the graph of the function.g1x2...
 4.5.20: Determine the vertical asymptotes of the graph of the function.f 1x...
 4.5.21: Determine the horizontal asymptote of the graph of the function.f 1...
 4.5.22: Determine the horizontal asymptote of the graph of the function.g1x...
 4.5.23: Determine the horizontal asymptote of the graph of the function.h1x...
 4.5.24: Determine the horizontal asymptote of the graph of the function.f 1...
 4.5.25: Determine the horizontal asymptote of the graph of the function.g1x...
 4.5.26: Determine the horizontal asymptote of the graph of the function.h1x...
 4.5.27: Determine the oblique asymptote of the graph of the function.g1x2 =...
 4.5.28: Determine the oblique asymptote of the graph of the functionf 1x2 =...
 4.5.29: Determine the oblique asymptote of the graph of the function.Determ...
 4.5.30: Determine the oblique asymptote of the graph of the function.g1x2 =...
 4.5.31: Determine the oblique asymptote of the graph of the function.f 1x2 ...
 4.5.32: Determine the oblique asymptote of the graph of the function.h1x2 =...
 4.5.33: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.34: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.35: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.36: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.37: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.38: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.39: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.40: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.41: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.42: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.43: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.44: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.45: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.46: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.47: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.48: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.49: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.50: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.51: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.52: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.53: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.54: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.55: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.56: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.57: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.58: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.59: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.60: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.61: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.62: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.63: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.64: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.65: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.66: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.67: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.68: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.69: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.70: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.71: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.72: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.73: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.74: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.75: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.76: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.77: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.78: Make a handdrawn graph for each of Exercises 3378. Be sure to labe...
 4.5.79: Find a rational function that satisfies the given conditions. Answe...
 4.5.80: Find a rational function that satisfies the given conditions. Answe...
 4.5.81: Find a rational function that satisfies the given conditions. Answe...
 4.5.82: Find a rational function that satisfies the given conditions. Answe...
 4.5.83: Medical Dosage. The function N1t2 = 0.8t + 1000 5t + 4 , t 15, give...
 4.5.84: Average Cost. The average cost per disc, in dollars, for a company ...
 4.5.85: Population Growth. The population P, in thousands, of a senior comm...
 4.5.86: Minimizing Surface Area. The HoldIt Container Co. is designing an ...
 4.5.87: A function is a correspondence between a first set, called the , an...
 4.5.88: The of a line containing 1x1, y12 and 1x2, y22 is given by 1 y2  y...
 4.5.89: The of the line with slope m and yintercept 10, b2 is y = mx + b.
 4.5.90: The of the line with slope m passing through 1x1, y12 is y  y1 = m...
 4.5.91: A(n) is a point 1a, 02.
 4.5.92: For each x in the domain of an odd function f, .
 4.5.93: are given by equations of the type x = a.
 4.5.94: The is a x1 + x2 2 , y1 + y2 2 b.
 4.5.95: A(n) is a point 10, b2.
 4.5.96: Graph y1 = x3 + 4 x and y2 = x2 using the same viewing window. Expl...
 4.5.97: Find the nonlinear asymptote of the function.f 1x2 = x5 + 2x3 + 4x2...
 4.5.98: Find the nonlinear asymptote of the function.f 1x2 = x4 + 3x2x2 + 1
 4.5.99: Graph the function.f 1x2 = 2x3 + x2  8x  4x3 + x2  9x  9
 4.5.100: Graph the function.f 1x2 = x3 + 4x2 + x  6x2  x  2
Solutions for Chapter 4.5: Rational Functions
Full solutions for College Algebra: Graphs and Models  5th Edition
ISBN: 9780321783950
Solutions for Chapter 4.5: Rational Functions
Get Full SolutionsSince 100 problems in chapter 4.5: Rational Functions have been answered, more than 28923 students have viewed full stepbystep solutions from this chapter. Chapter 4.5: Rational Functions includes 100 full stepbystep solutions. This textbook survival guide was created for the textbook: College Algebra: Graphs and Models, edition: 5. College Algebra: Graphs and Models was written by and is associated to the ISBN: 9780321783950. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

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).

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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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 .

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).