 6.1.1: A rational expression consists of a/an __________________ divided b...
 6.1.2: The domain of a rational function is the set of all real numbers ex...
 6.1.3: A vertical line that the graph of a function approaches, but does n...
 6.1.4: A horizontal line that the graph of a rational function approaches ...
 6.1.5: We simplify a rational expression by __________________ the numerat...
 6.1.6: The rational expression (x + 3)(x + 5) x + 3 simplifies to ________...
 6.1.7: True or false: x2  9 9 simplifies to x2  1. __________________
 6.1.8: The rational expression x  5 5  x simplifies to __________________.
 6.1.9: The product of two rational expressions is the product of their ___...
 6.1.10: The quotient of two rational expressions is the product of the firs...
 6.1.11: x 7 # x 10 =
 6.1.12: x 7 , x 10 =
 6.1.13: f(x) = 3x x2  8x + 15
 6.1.14: f(x) = 3x x2  13x + 36
 6.1.15: f(x) = (x  1) 2 3x2  2x  8
 6.1.16: f(x) = (x  1) 2 4x2  13x + 3
 6.1.17: Find f(4).
 6.1.18: Find f(1).
 6.1.19: What is the domain of f ? What is the range of f ?
 6.1.20: What are the equations of the vertical asymptotes of the graph of f ?
 6.1.21: Describe the end behavior of the graph at the far left. What is the...
 6.1.22: Describe the end behavior of the graph at the far right. What is th...
 6.1.23: Explain how the graph shows that f(2) does not exist.
 6.1.24: Explain how the graph shows that f(2) does not exist.
 6.1.25: How can you tell that this is not the graph of a polynomial function?
 6.1.26: List two real numbers that are not function values of f
 6.1.27: x2  4 x  2
 6.1.28: x2  25 x  5
 6.1.29: x + 2 x2  x  6
 6.1.30: x + 1 x2  2x  3
 6.1.31: 4x + 20 x2 + 5x
 6.1.32: x + 1 x2  2x  3
 6.1.33: 4y  20 y2  25
 6.1.34: 6y  42 y2  49
 6.1.35: 3x  5 25  9x2
 6.1.36: 5x  2 4  25x2
 6.1.37: y2  49 y2  14y + 49
 6.1.38: y2  9 y2  6y + 9
 6.1.39: x2 + 7x  18 x2  3x + 2
 6.1.40: x2  4x  5 x2 + 5x + 4
 6.1.41: 3x + 7 3x + 10
 6.1.42: 2x + 3 2x + 5
 6.1.43: x2  x  12 16  x2
 6.1.44: x2  7x + 12 9  x2
 6.1.45: x2 + 3xy  10y2 3x2  7xy + 2y2
 6.1.46: x2 + 2xy  3y2 2x2 + 5xy  3y2
 6.1.47: x3  8 x2  4
 6.1.48: x3  1 x2  1
 6.1.49: x3 + 4x2  3x  12 x + 4
 6.1.50: x3  2x2 + x  2 x  2
 6.1.51: x  3 x + 7 # 3x + 21 2x  6
 6.1.52: x  2 x + 3 # 2x + 6 5x  10
 6.1.53: x2  49 x2  4x  21 # x + 3 x
 6.1.54: x2  25 x2  3x  10 # x + 2 x
 6.1.55: x2  9 x2  x  6 # x2 + 5x + 6 x2 + x  6
 6.1.56: x2  1 x2  4 # x2  5x + 6 x2  2x  3
 6.1.57: x2 + 4x + 4 x2 + 8x + 16 # (x + 4) 3 (x + 2) 3
 6.1.58: x2  2x + 1 x2  4x + 4 # (x  2) 3 (x  1) 3
 6.1.59: 8y + 2 y2  9 # 3  y 4y2 + y
 6.1.60: 6y + 2 y2  1 # 1  y 3y2 + y
 6.1.61: y3  8 y2  4 # y + 2 2y
 6.1.62: y2 + 6y + 9 y3 + 27 # 1 y + 3
 6.1.63: (x  3) # x2 + x + 1 x2  5x + 6
 6.1.64: (x + 1) # x + 2 x2 + 7x + 6
 6.1.65: x2 + xy x2  y2 # 4x  4y x
 6.1.66: x2  y2 x # x2 + xy x + y
 6.1.67: x2 + 2xy + y2 x2  2xy + y2 # 4x  4y 3x + 3y
 6.1.68: 2x2  3xy  2y2 3x2  4xy + y2 # 3x2  2xy  y2 x2 + xy  6y2
 6.1.69: 4a2 + 2ab + b2 2a + b # 4a2  b2 8a3  b3
 6.1.70: 27a3  8b3 b2  b  6 # bc  b  3c + 3 3ac  2bc  3a + 2b
 6.1.71: 10z2 + 13z  3 3z2  8z + 5 # 2z2  3z  2z + 3 25z2  10z + 1 # 15...
 6.1.72: 2z2  2z  12 z2  49 # 4z2  1 2z2 + 5z + 2 # 2z2  13z  7 2z2  ...
 6.1.73: x + 5 7 , 4x + 20 9
 6.1.74: x + 1 3 , 3x + 3 7
 6.1.75: 4 y  6 , 40 7y  42
 6.1.76: 7 y  5 , 28 3y  15
 6.1.77: x2  2x 15 , x  2 5
 6.1.78: x2  x 15 , x  1 5
 6.1.79: y2  25 2y  2 , y2 + 10y + 25 y2 + 4y  5
 6.1.80: y2 + y y2  4 , y2  1 y2 + 5y + 6
 6.1.81: (x2  16) , x2 + 3x  4 x2 + 4
 6.1.82: (x2 + 4x  5) , x2  25 x + 7
 6.1.83: y2  4y  21 y2  10y + 25 , y2 + 2y  3 y2  6y + 5
 6.1.84: y2 + 4y  21 y2 + 3y  28 , y2 + 14y + 48 y2 + 4y  32
 6.1.85: 8x3  1 4x2 + 2x + 1 , x  1 (x  1) 2
 6.1.86: x2  9 x3  27 , x2 + 6x + 9 x2 + 3x + 9
 6.1.87: x2  4y2 x2 + 3xy + 2y2 , x2  4xy + 4y2 x + y
 6.1.88: xy  y2 x2 + 2x + 1 , 2x2 + xy  3y2 2x2 + 5xy + 3y2
 6.1.89: x4  y8 x2 + y4 , x2  y4 3x2
 6.1.90: (x  y) 3 x3  y3 , x2  2xy + y2 x2  y2
 6.1.91: x3  4x2 + x  4 2x3  8x2 + x  4 # 2x3 + 2x2 + x + 1 x4  x3 + x2...
 6.1.92: y3 + y2 + yz2 + z2 y3 + y + y2 + 1 # y3 + y + y2z + z 2y2 + 2yz  y...
 6.1.93: ax  ay + 3x  3y x3 + y3 , ab + 3b + ac + 3c xy  x2  y2
 6.1.94: a3 + b3 ac  ad  bc + bd , ab  a2  b2 ac  ad + bc  bd
 6.1.95: a2b + b 3a2  4a  20 # a2 + 5a 2a2 + 11a + 5 , ab2 6a2  17a  10
 6.1.96: a2  8a + 15 2a3  10a2 # 2a2 + 3a 3a3  27a , 14a + 21 a2  6a  27
 6.1.97: a  b 4c , b  a c , a  b c 2
 6.1.98: a a  b 4c , b  a c b , a  b c 2
 6.1.99: f(x) = 7x  4
 6.1.100: f(x) = 3x + 5
 6.1.101: f(x) = x2  5x + 3
 6.1.102: f(x) = 3x2  4x + 7
 6.1.103: Find a f g b(x) and the domain of f g .
 6.1.104: Find a g f b(x) and the domain of g f .
 6.1.105: Find and interpret f(60). Identify your solution as a point on the ...
 6.1.106: Find and interpret f(80). Identify your solution as a point on the ...
 6.1.107: What value of x must be excluded from the rational functions domain...
 6.1.108: What happens to the cost as x approaches 100%? How is this shown by...
 6.1.109: After eating sugar, when is the pH level the lowest? Use the functi...
 6.1.110: Use the graph to obtain a reasonable estimate, to the nearest tenth...
 6.1.111: According to the graph, what is the normal pH level of the human mo...
 6.1.112: Use the graph to describe what happens to the pH level during the f...
 6.1.113: Find P(10). Describe what this means in terms of the incidence rati...
 6.1.114: Find P(9). Round to the nearest percent. Describe what this means i...
 6.1.115: What is the horizontal asymptote of the graph? Describe what this m...
 6.1.116: According to the model and its graph, is there a disease for which ...
 6.1.117: What is a rational expression? Give an example with your explanation.
 6.1.118: What is a rational function? Provide an example.
 6.1.119: What is the domain of a rational function?
 6.1.120: If you are given the equation of a rational function, explain how t...
 6.1.121: Describe two ways the graph of a rational function differs from the...
 6.1.122: What is a vertical asymptote?
 6.1.123: What is a horizontal asymptote?
 6.1.124: Explain how to simplify a rational expression.
 6.1.125: Explain how to simplify a rational expression with opposite factors...
 6.1.126: Explain how to multiply rational expressions.
 6.1.127: Explain how to divide rational expressions.
 6.1.128: Although your friend has a family history of heart disease, he smok...
 6.1.129: x2 + x 3x # 6x x + 1 = 2x
 6.1.130: x3  25x x2  3x  10 # x + 2 x = x + 5
 6.1.131: x2  9 x + 4 , x  3 x + 4 = x  3
 6.1.132: (x  5) , 2x2  11x + 5 4x2  1 = 2x  1
 6.1.133: Use the TABLE feature of a graphing utility to verify the domains t...
 6.1.134: a. Graph f(x) = x2  x  2 x  2 and g(x) = x + 1 in the same viewi...
 6.1.135: I cannot simplify, multiply, or divide rational expressions without...
 6.1.136: I simplified 2(x + 2)  5(x + 1) (x + 2)(x + 1) by dividing the num...
 6.1.137: The values to exclude from the domain of f(x) = x  3 x  7 are 3 a...
 6.1.138: When performing the division 3x x + 2 , (x + 2) 2 x  4 , I began b...
 6.1.139: x2  25 x  5 = x  5
 6.1.140: x2 + 7 7 = x2 + 1
 6.1.141: The domain of f(x) = 7 x(x  3) + 5(x  3) is ( , 3) (3, ).
 6.1.142: The restrictions on the values of x when performing the division f(...
 6.1.143: Graph: f(x) = x2  x  2 x  2 .
 6.1.144: Simplify: 6x3n + 6x2nyn x2n  y2n .
 6.1.145: Divide: y2n  1 y2n + 3yn + 2 , y2n + yn  12 y2n  yn  6 .
 6.1.146: Solve for x in terms of a and write the resulting rational expressi...
 6.1.147: Graph: 4x  5y 20. (Section 4.4, Example 1)
 6.1.148: Multiply: (2x  5)(x2  3x  6). (Section 5.2, Example 3)
 6.1.149: Simplify: ab 3c 4 4a5b10c 3 2 . (Section 1.6, Example 9)
 6.1.150: 7 10  3 10
 6.1.151: 1 2 + 2 3
 6.1.152: 7 15  3 10
Solutions for Chapter 6.1: Rational Expressions and Functions: Multiplying and Dividing
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 6.1: Rational Expressions and Functions: Multiplying and Dividing
Get Full SolutionsIntermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. Since 152 problems in chapter 6.1: Rational Expressions and Functions: Multiplying and Dividing have been answered, more than 45412 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. Chapter 6.1: Rational Expressions and Functions: Multiplying and Dividing includes 152 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

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.)

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

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.

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.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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·