5.1) Is a onetoone function?
5.2) The graph of a function f with domain is shown in the figure. Sketch the graph of .
5.3) Exer. 34: (a) Find . (b) Sketch the graphs of f and on the same coordinate plane
5.4) Exer. 34: (a) Find . (b) Sketch the graphs of f and on the same coordinate plane
5.5) Refer to the figure to determine each of the following
5.6) Suppose f and g are onetoone functions such that , , and . Find the value, if possible.
5.7) Exer. 722: Sketch the graph of f.
5.8) Exer. 722: Sketch the graph of f.
5.9) Exer. 722: Sketch the graph of f.
5.10) Exer. 722: Sketch the graph of f.
5.11) Exer. 722: Sketch the graph of f.
5.12) Exer. 722: Sketch the graph of f.
5.13) Exer. 722: Sketch the graph of f.
5.14) Exer. 722: Sketch the graph of f.
5.15) Exer. 722: Sketch the graph of f.
5.16) Exer. 722: Sketch the graph of f.
5.17) Exer. 722: Sketch the graph of f.
5.18) Exer. 722: Sketch the graph of f.
5.19) Exer. 722: Sketch the graph of f.
5.20) Exer. 722: Sketch the graph of f.
5.21) Exer. 722: Sketch the graph of f.
5.22) Exer. 722: Sketch the graph of f.
5.23) Exer. 2324: Evaluate without using a calculator.
5.24) Exer. 2324: Evaluate without using a calculator.
5.25) Exer. 2544: Solve the equation without using a calculator.
5.26) Exer. 2544: Solve the equation without using a calculator.
5.27) Exer. 2544: Solve the equation without using a calculator.
5.28) Exer. 2544: Solve the equation without using a calculator.
5.29) Exer. 2544: Solve the equation without using a calculator.
5.30) Exer. 2544: Solve the equation without using a calculator.
5.31) Exer. 2544: Solve the equation without using a calculator.
5.32) Exer. 2544: Solve the equation without using a calculator.
5.33) Exer. 2544: Solve the equation without using a calculator.
5.34) Exer. 2544: Solve the equation without using a calculator.
5.35) Exer. 2544: Solve the equation without using a calculator.
5.36) Exer. 2544: Solve the equation without using a calculator.
5.37) Exer. 2544: Solve the equation without using a calculator.
5.38) Exer. 2544: Solve the equation without using a calculator.
5.39) Exer. 2544: Solve the equation without using a calculator.
5.40) Exer. 2544: Solve the equation without using a calculator.
5.41) Exer. 2544: Solve the equation without using a calculator.
5.42) Exer. 2544: Solve the equation without using a calculator.
5.43) Exer. 2544: Solve the equation without using a calculator.
5.44) Exer. 2544: Solve the equation without using a calculator.
5.45) Express in terms of logarithms of x, y, and z.
5.46) Express as one logarithm
5.47) Find an exponential function that has yintercept 6 and passes through the point (1, 8).
5.49) Exer. 4950: Use common logarithms to solve the equation for x in terms of y.
5.50) Exer. 4950: Use common logarithms to solve the equation for x in terms of y.
5.51) Exer. 5152: Approximate x to three significant figures.
5.52) Exer. 5152: Approximate x to three significant figures.
Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. This expansive textbook survival guide covers the following chapters and their solutions. Since 76 problems in chapter 5: INVERSE, EXPONENTIAL, AND LOGARITHMIC FUNCTIONS have been answered, more than 176846 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12. Chapter 5: INVERSE, EXPONENTIAL, AND LOGARITHMIC FUNCTIONS includes 76 full stepbystep solutions.

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!

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Solvable system Ax = b.
The right side b is in the column space of A.