 6.1.1: Answers are given at the end of these exercises. If you get a wrong...
 6.1.2: Answers are given at the end of these exercises. If you get a wrong...
 6.1.3: Answers are given at the end of these exercises. If you get a wrong...
 6.1.4: Given two functions f and g, the , denoted , is defined by
 6.1.5: f1g1x22 = f1x2 # g(x).
 6.1.6: The domain of the composite function 1f g21x2 is the same as the do...
 6.1.7: In 7 and 8, evaluate each expression using the values given in the ...
 6.1.8: In 7 and 8, evaluate each expression using the values given in the ...
 6.1.9: In 9 and 10, evaluate each expression using the graphs of and shown...
 6.1.10: In 9 and 10, evaluate each expression using the graphs of and shown...
 6.1.11: In 1120, for the given functions and , find: (a) (b) (c) (d) 1f g21...
 6.1.12: In 1120, for the given functions and , find: (a) (b) (c) (d) 1f g21...
 6.1.13: In 1120, for the given functions and , find: (a) (b) (c) (d) 1f g21...
 6.1.14: In 1120, for the given functions and , find: (a) (b) (c) (d) 1f g21...
 6.1.15: In 1120, for the given functions and , find: (a) (b) (c) (d) 1f g21...
 6.1.16: In 1120, for the given functions and , find: (a) (b) (c) (d) 1f g21...
 6.1.17: In 1120, for the given functions and , find: (a) (b) (c) (d) 1f g21...
 6.1.18: In 1120, for the given functions and , find: (a) (b) (c) (d) 1f g21...
 6.1.19: In 1120, for the given functions and , find: (a) (b) (c) (d) 1f g21...
 6.1.20: In 1120, for the given functions and , find: (a) (b) (c) (d) 1f g21...
 6.1.21: In 2128, find the domain of the composite function f g.
 6.1.22: In 2128, find the domain of the composite function f g.
 6.1.23: In 2128, find the domain of the composite function f g.
 6.1.24: In 2128, find the domain of the composite function f g.
 6.1.25: In 2128, find the domain of the composite function f g.
 6.1.26: In 2128, find the domain of the composite function f g.
 6.1.27: In 2128, find the domain of the composite function f g.
 6.1.28: In 2128, find the domain of the composite function f g.
 6.1.29: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.30: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.31: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.32: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.33: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.34: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.35: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.36: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.37: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.38: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.39: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.40: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.41: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.42: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.43: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.44: In 2944, for the given functions and find: (a) (b) (c) (d) State th...
 6.1.45: In 4552, show that 1f g21x2 = 1g f21x2 = x.
 6.1.46: In 4552, show that 1f g21x2 = 1g f21x2 = x.
 6.1.47: In 4552, show that 1f g21x2 = 1g f21x2 = x.
 6.1.48: In 4552, show that 1f g21x2 = 1g f21x2 = x.
 6.1.49: In 4552, show that 1f g21x2 = 1g f21x2 = x.
 6.1.50: In 4552, show that 1f g21x2 = 1g f21x2 = x.
 6.1.51: In 4552, show that 1f g21x2 = 1g f21x2 = x.
 6.1.52: In 4552, show that 1f g21x2 = 1g f21x2 = x.
 6.1.53: In 5358, find functions and so that f g f g = H.
 6.1.54: In 5358, find functions and so that f g f g = H.
 6.1.55: In 5358, find functions and so that f g f g = H.
 6.1.56: In 5358, find functions and so that f g f g = H.
 6.1.57: In 5358, find functions and so that f g f g = H.
 6.1.58: In 5358, find functions and so that f g f g = H.
 6.1.59: If and find and f1x2 = 2x g1x2 = 2, 1f g21x2 3  3x2 + 4x  1
 6.1.60: If f1x2 = 1f f21x2. x + find
 6.1.61: If and find a so that the graph of crosses the yaxis at 23
 6.1.62: If and find a so that the graph of crosses the yaxis at 68.
 6.1.63: In 63 and 64, use the functions f and g to find: (a) (b) (c) the do...
 6.1.64: In 63 and 64, use the functions f and g to find: (a) (b) (c) the do...
 6.1.65: The surface area S (in square meters) of a hotair balloon is given...
 6.1.66: The volume V (in cubic meters) of the hotair balloon described in ...
 6.1.67: The number N of cars produced at a certain factory in one day after...
 6.1.68: The spread of oil leaking from a tanker is in the shape of a circle...
 6.1.69: The price p, in dollars, of a certain product and the quantity x so...
 6.1.70: The price p, in dollars, of a certain commodity and the quantity x ...
 6.1.71: The volume V of a right circular cylinder of height h and radius r ...
 6.1.72: The volume V of a right circular cone is If the height is twice the...
 6.1.73: Traders often buy foreign currency in hope of making money when the...
 6.1.74: The function converts a temperature in degrees Fahrenheit, F, to a ...
 6.1.75: The manufacturer of a computer is offering two discounts on last ye...
 6.1.76: If and are odd functions, show that the composite function is also ...
 6.1.77: If is an odd function and is an even function, show that the compos...
Solutions for Chapter 6.1: Composite Functions
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter 6.1: Composite Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.1: Composite Functions includes 77 full stepbystep solutions. Since 77 problems in chapter 6.1: Composite Functions have been answered, more than 36683 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9780321716811. This textbook survival guide was created for the textbook: College Algebra, edition: 9.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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!

Column space C (A) =
space of all combinations of the columns of A.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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).

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

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.

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(DÂ» O.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

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.

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.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.