- 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
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Invert A by row operations on [A I] to reach [I A-I].
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
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).
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.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
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.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
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.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
Having trouble accessing your account? Let us help you, contact support at +1(510) 944-1054 or firstname.lastname@example.org
Forgot password? Reset it here