 2.3.1: We exclude from a functions domain real numbers that cause division...
 2.3.2: We exclude from a functions domain real numbers that result in a sq...
 2.3.3: (f + g)(x) = ______________
 2.3.4: (f  g)(x) = ______________
 2.3.5: (fg)(x) = ______________
 2.3.6: f g (x) = ______________, provided ______________ 0
 2.3.7: The domain of f(x) = 5x + 7 consists of all real numbers, represent...
 2.3.8: The domain of g(x) = 3 x  2 consists of all real numbers except 2,...
 2.3.9: The domain of h(x) = 1 x + 7 x  3 consists of all real numbers exc...
 2.3.10: f(x) = 1 x + 8 + 3 x  10
 2.3.11: f(x) = 3x + 1, g(x) = 2x  6
 2.3.12: f(x) = 4x + 2, g(x) = 2x  9
 2.3.13: f(x) = x  5, g(x) = 3x2
 2.3.14: f(x) = x  6, g(x) = 2x2
 2.3.15: f(x) = 2x2  x  3, g(x) = x + 1
 2.3.16: f(x) = 4x2  x  3, g(x) = x + 1
 2.3.17: Let f(x) = 5x and g(x) = 2x  3. Find (f + g)(x), (f  g)(x), (fg)...
 2.3.18: Let f(x) = 4x and g(x) = 3x + 5. Find (f + g)(x), (f  g)(x), (fg...
 2.3.19: f(x) = 3x + 7, g(x) = 9x + 10
 2.3.20: f(x) = 7x + 4, g(x) = 5x  2
 2.3.21: f(x) = 3x + 7, g(x) = 2 x  5
 2.3.22: f(x) = 7x + 4, g(x) = 2 x  6
 2.3.23: f(x) = 1 x , g(x) = 2 x  5
 2.3.24: f(x) = 1 x , g(x) = 2 x  6
 2.3.25: f(x) = 8x x  2 , g(x) = 6 x + 3
 2.3.26: f(x) = 9x x  4 , g(x) = 7 x + 8
 2.3.27: f(x) = 8x x  2 , g(x) = 6 2  x
 2.3.28: f(x) = 9x x  4 , g(x) = 7 4  x
 2.3.29: f(x) = x2 , g(x) = x3
 2.3.30: f(x) = x2 + 1, g(x) = x3  1
 2.3.31: (f + g)(x) and (f + g)(3)
 2.3.32: (f + g)(x) and (f + g)(4)
 2.3.33: f(2) + g(2)
 2.3.34: f(3) + g(3)
 2.3.35: (f  g)(x) and (f  g)(5)
 2.3.36: (f  g)(x) and (f  g)(6)
 2.3.37: f(2)  g(2)
 2.3.38: f(3)  g(3)
 2.3.39: (fg)(2)
 2.3.40: (fg)(3)
 2.3.41: (fg)(5)
 2.3.42: (fg)(6)
 2.3.43: a f g b(x) and a f g b(1)
 2.3.44: a f g b(x) and a f g b(3)
 2.3.45: a f g b(1)
 2.3.46: a f g b(0)
 2.3.47: The domain of f + g
 2.3.48: The domain of f  g
 2.3.49: The domain of f g
 2.3.50: The domain of fg
 2.3.51: Find (f + g)(3).
 2.3.52: Find (g  f)(2).
 2.3.53: Find (fg)(2).
 2.3.54: Find a g f b(3).
 2.3.55: Find the domain of f + g.
 2.3.56: Find the domain of f g .
 2.3.57: Graph f + g.
 2.3.58: Graph f  g.
 2.3.59: Find (f + g)(1)  (g  f)(1).
 2.3.60: Find (f + g)(1)  (g  f)(0).
 2.3.61: Find (fg)(2)  c a f g b(1)d 2 .
 2.3.62: Find (fg)(2)  c a g f b(0)d 2 .
 2.3.63: a. Write a function that models the total U.S. population for the y...
 2.3.64: a. Write a function that models the difference between the female U...
 2.3.65: a. Write a function that models the ratio of men to women in the U....
 2.3.66: A company that sells radios has yearly fixed costs of $600,000. It ...
 2.3.67: If a function is defined by an equation, explain how to find its do...
 2.3.68: If equations for functions f and g are given, explain how to find f...
 2.3.69: If the equations of two functions are given, explain how to obtain ...
 2.3.70: If equations for functions f and g are given, describe two ways to ...
 2.3.71: y1 = 2x + 3 y2 = 2  2x y3 = y1 + y2
 2.3.72: y1 = x  4 y2 = 2x y3 = y1  y2
 2.3.73: y1 = x y2 = x  4 y3 = y1 # y2
 2.3.74: y1 = x2  2x y2 = x y3 = y1 y2
 2.3.75: In Exercise 74, use the TRACE feature to trace along y3 . What happ...
 2.3.76: There is an endless list of real numb
 2.3.77: I used a function to model data from 1980 through 2005. The indepen...
 2.3.78: If I have equations for functions f and g, and 3 is in both domains...
 2.3.79: I have two functions. Function f models total world population x ye...
 2.3.80: If (f + g)(a) = 0, then f(a) and g(a) must be opposites, or additiv...
 2.3.81: If (f  g)(a) = 0, then f(a) and g(a) must be equal.
 2.3.82: If a f g b(a) = 0, then f(a) must be 0.
 2.3.83: If (fg)(a) = 0, then f(a) must be 0.
 2.3.84: Solve for b: R = 3(a + b). (Section 1.5, Example 6)
 2.3.85: Solve: 3(6  x) = 3  2(x  4). (Section 1.4, Example 3)
 2.3.86: If f(x) = 6x  4, find f(b + 2). (Section 2.1, Example 3)
 2.3.87: Consider 4x  3y = 6. a. What is the value of x when y = 0? b. What...
 2.3.88: a. Graph y = 2x + 4. Select integers for x from 3 to 1, inclusive....
 2.3.89: Solve for y: 5x + 3y = 12.
Solutions for Chapter 2.3: The Algebra of Functions
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 2.3: The Algebra of Functions
Get Full SolutionsSince 89 problems in chapter 2.3: The Algebra of Functions have been answered, more than 7397 students have viewed full stepbystep solutions from this chapter. Chapter 2.3: The Algebra of Functions includes 89 full stepbystep solutions. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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).

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
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