 3.4.1: How does one find the domain of the quotient of two functions, f _ g ?
 3.4.2: What is the composition of two functions, f g ?
 3.4.3: If the order is reversed when composing two functions, can the resu...
 3.4.4: How do you find the domain for the composition of two functions, f g ?
 3.4.5: For the following exercises, determine the domain for each function...
 3.4.6: For the following exercises, determine the domain for each function...
 3.4.7: For the following exercises, determine the domain for each function...
 3.4.8: For the following exercises, determine the domain for each function...
 3.4.9: For the following exercises, determine the domain for each function...
 3.4.10: For the following exercises, determine the domain for each function...
 3.4.11: For the following exercise, find the indicated function given f(x) ...
 3.4.12: For the following exercises, use each pair of functions to find f(g...
 3.4.13: For the following exercises, use each pair of functions to find f(g...
 3.4.14: For the following exercises, use each pair of functions to find f(g...
 3.4.15: For the following exercises, use each pair of functions to find f(g...
 3.4.16: For the following exercises, use each pair of functions to find f(g...
 3.4.17: For the following exercises, use each pair of functions to find f(g...
 3.4.18: For the following exercises, use each set of functions to find f(g(...
 3.4.19: For the following exercises, use each set of functions to find f(g(...
 3.4.20: Given f(x) = _ 1 x , and g(x) = x 3, find the following: a. (f g)(x...
 3.4.21: Given f(x) = 2 4x and g(x) = _ 3 x , find the following: a. ( g f )...
 3.4.22: Given the functions f(x) = _ 1 x x and g(x) = _ 1 1 + x2 , find the...
 3.4.23: Given functions p(x) = _ 1 x and m(x) = x 2 4, state the domain of ...
 3.4.24: Given functions q(x) = _ 1 x and h(x) = x2 9, state the domain of e...
 3.4.25: For f(x) = _ 1 x and g(x) = x 1 , write the domain of (f g)(x) in i...
 3.4.26: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.27: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.28: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.29: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.30: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.31: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.32: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.33: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.34: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.35: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.36: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.37: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.38: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.39: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.40: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.41: For the following exercises, find functions f(x) and g(x) so the gi...
 3.4.42: For the following exercises, use the graphs of f, shown in Figure 4...
 3.4.43: For the following exercises, use the graphs of f, shown in Figure 4...
 3.4.44: For the following exercises, use the graphs of f, shown in Figure 4...
 3.4.45: For the following exercises, use the graphs of f, shown in Figure 4...
 3.4.46: For the following exercises, use the graphs of f, shown in Figure 4...
 3.4.47: For the following exercises, use the graphs of f, shown in Figure 4...
 3.4.48: For the following exercises, use the graphs of f, shown in Figure 4...
 3.4.49: For the following exercises, use the graphs of f, shown in Figure 4...
 3.4.50: For the following exercises, use graphs of f(x), shown in Figure 6,...
 3.4.51: For the following exercises, use graphs of f(x), shown in Figure 6,...
 3.4.52: For the following exercises, use graphs of f(x), shown in Figure 6,...
 3.4.53: For the following exercises, use graphs of f(x), shown in Figure 6,...
 3.4.54: For the following exercises, use graphs of f(x), shown in Figure 6,...
 3.4.55: For the following exercises, use graphs of f(x), shown in Figure 6,...
 3.4.56: For the following exercises, use graphs of f(x), shown in Figure 6,...
 3.4.57: For the following exercises, use graphs of f(x), shown in Figure 6,...
 3.4.58: For the following exercises, use the function values for f and g sh...
 3.4.59: For the following exercises, use the function values for f and g sh...
 3.4.60: For the following exercises, use the function values for f and g sh...
 3.4.61: For the following exercises, use the function values for f and g sh...
 3.4.62: For the following exercises, use the function values for f and g sh...
 3.4.63: For the following exercises, use the function values for f and g sh...
 3.4.64: For the following exercises, use the function values for f and g sh...
 3.4.65: For the following exercises, use the function values for f and g sh...
 3.4.66: For the following exercises, use the function values for f and g sh...
 3.4.67: For the following exercises, use the function values for f and g sh...
 3.4.68: For the following exercises, use the function values for f and g sh...
 3.4.69: For the following exercises, use the function values for f and g sh...
 3.4.70: For the following exercises, use the function values for f and g sh...
 3.4.71: For the following exercises, use the function values for f and g sh...
 3.4.72: For the following exercises, use each pair of functions to find f(g...
 3.4.73: For the following exercises, use each pair of functions to find f(g...
 3.4.74: For the following exercises, use each pair of functions to find f(g...
 3.4.75: For the following exercises, use each pair of functions to find f(g...
 3.4.76: For the following exercises, use the functions f(x) = 2x2 + 1 and g...
 3.4.77: For the following exercises, use the functions f(x) = 2x2 + 1 and g...
 3.4.78: For the following exercises, use the functions f(x) = 2x2 + 1 and g...
 3.4.79: For the following exercises, use the functions f(x) = 2x2 + 1 and g...
 3.4.80: For the following exercises, use f(x) = x3 + 1 and g(x) = 3 x 1 . F...
 3.4.81: For the following exercises, use f(x) = x3 + 1 and g(x) = 3 x 1 . F...
 3.4.82: For the following exercises, use f(x) = x3 + 1 and g(x) = 3 x 1 . W...
 3.4.83: For the following exercises, use f(x) = x3 + 1 and g(x) = 3 x 1 . W...
 3.4.84: For the following exercises, use f(x) = x3 + 1 and g(x) = 3 x 1 . L...
 3.4.85: For the following exercises, let F(x) = (x + 1)5 , f(x) = x5 , and ...
 3.4.86: For the following exercises, let F(x) = (x + 1)5 , f(x) = x5 , and ...
 3.4.87: For the following exercises, find the composition when f(x) = x2 + ...
 3.4.88: For the following exercises, find the composition when f(x) = x2 + ...
 3.4.89: For the following exercises, find the composition when f(x) = x2 + ...
 3.4.90: The function D(p) gives the number of items that will be demanded w...
 3.4.91: The function A(d) gives the pain level on a scale of 0 to 10 experi...
 3.4.92: A store offers customers a 30% discount on the price x of selected ...
 3.4.93: A rain drop hitting a lake makes a circular ripple. If the radius, ...
 3.4.94: A forest fire leaves behind an area of grass burned in an expanding...
 3.4.95: Use the function you found in the previous exercise to find the tot...
 3.4.96: The radius r, in inches, of a spherical balloon is related to the v...
 3.4.97: The number of bacteria in a refrigerated food product is given by N...
Solutions for Chapter 3.4: COMPOSITION OF FUNCTIONS
Full solutions for College Algebra  1st Edition
ISBN: 9781938168383
Solutions for Chapter 3.4: COMPOSITION OF FUNCTIONS
Get Full SolutionsChapter 3.4: COMPOSITION OF FUNCTIONS includes 97 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 97 problems in chapter 3.4: COMPOSITION OF FUNCTIONS have been answered, more than 20872 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra, edition: 1. College Algebra was written by and is associated to the ISBN: 9781938168383.

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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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.

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

Pascal matrix
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).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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.

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.