- 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
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
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.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Gram-Schmidt 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.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
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.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
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).
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
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.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.