 1.3.1: A relation that assigns to each element x from a set of inputs, or ...
 1.3.2: For an equation that represents y as a function of x, the _______ v...
 1.3.3: Can the ordered pairs (3, 0) and (3, 5) represent a function?
 1.3.4: To find g(x + 1), what do you substitute for x in the function g(x)...
 1.3.5: Does the domain of the function f(x) = 1 + x include x = 2?
 1.3.6: Is the domain of a piecewisedefined function implied or explicitly...
 1.3.7: In Exercises 710, does the relation describe a function? Explain yo...
 1.3.8: In Exercises 710, does the relation describe a function? Explain yo...
 1.3.9: In Exercises 710, does the relation describe a function? Explain yo...
 1.3.10: In Exercises 710, does the relation describe a function? Explain yo...
 1.3.11: In Exercises 1114, determine whether the relation represents y as a...
 1.3.12: In Exercises 1114, determine whether the relation represents y as a...
 1.3.13: In Exercises 1114, determine whether the relation represents y as a...
 1.3.14: In Exercises 1114, determine whether the relation represents y as a...
 1.3.15: In Exercises 15 and 16, which sets of ordered pairs represent funct...
 1.3.16: In Exercises 15 and 16, which sets of ordered pairs represent funct...
 1.3.17: In Exercises 17 and 18, use the graph, which shows the average pric...
 1.3.18: In Exercises 17 and 18, use the graph, which shows the average pric...
 1.3.19: In Exercises 1930, determine whether the equation represents y as a...
 1.3.20: In Exercises 1930, determine whether the equation represents y as a...
 1.3.21: In Exercises 1930, determine whether the equation represents y as a...
 1.3.22: In Exercises 1930, determine whether the equation represents y as a...
 1.3.23: In Exercises 1930, determine whether the equation represents y as a...
 1.3.24: In Exercises 1930, determine whether the equation represents y as a...
 1.3.25: In Exercises 1930, determine whether the equation represents y as a...
 1.3.26: In Exercises 1930, determine whether the equation represents y as a...
 1.3.27: In Exercises 1930, determine whether the equation represents y as a...
 1.3.28: In Exercises 1930, determine whether the equation represents y as a...
 1.3.29: In Exercises 1930, determine whether the equation represents y as a...
 1.3.30: In Exercises 1930, determine whether the equation represents y as a...
 1.3.31: In Exercises 3146, evaluate the function at each specified value of...
 1.3.32: In Exercises 3146, evaluate the function at each specified value of...
 1.3.33: In Exercises 3146, evaluate the function at each specified value of...
 1.3.34: In Exercises 3146, evaluate the function at each specified value of...
 1.3.35: In Exercises 3146, evaluate the function at each specified value of...
 1.3.36: In Exercises 3146, evaluate the function at each specified value of...
 1.3.37: In Exercises 3146, evaluate the function at each specified value of...
 1.3.38: In Exercises 3146, evaluate the function at each specified value of...
 1.3.39: In Exercises 3146, evaluate the function at each specified value of...
 1.3.40: In Exercises 3146, evaluate the function at each specified value of...
 1.3.41: In Exercises 3146, evaluate the function at each specified value of...
 1.3.42: In Exercises 3146, evaluate the function at each specified value of...
 1.3.43: In Exercises 3146, evaluate the function at each specified value of...
 1.3.44: In Exercises 3146, evaluate the function at each specified value of...
 1.3.45: In Exercises 3146, evaluate the function at each specified value of...
 1.3.46: In Exercises 3146, evaluate the function at each specified value of...
 1.3.47: In Exercises 4750, assume that the domain of f is the set A = {2, 1...
 1.3.48: In Exercises 4750, assume that the domain of f is the set A = {2, 1...
 1.3.49: In Exercises 4750, assume that the domain of f is the set A = {2, 1...
 1.3.50: In Exercises 4750, assume that the domain of f is the set A = {2, 1...
 1.3.51: In Exercises 51 and 52, complete the table. 51. h(t) = 1 2t + 3 t 5...
 1.3.52: In Exercises 51 and 52, complete the table. f(s) = s 2 s 2 s 0 1 3 2
 1.3.53: In Exercises 5356, find all values of x such that f(x) = 0 . f(x) =...
 1.3.54: In Exercises 5356, find all values of x such that f(x) = 0 . f(x) =...
 1.3.55: In Exercises 5356, find all values of x such that f(x) = 0 f(x) = 9x 4
 1.3.56: In Exercises 5356, find all values of x such that f(x) = 0 f(x) = 2...
 1.3.57: In Exercises 5766, find the domain of the function f(x) = 5x2 + 2x 1
 1.3.58: In Exercises 5766, find the domain of the function g(x) = 1 2x2
 1.3.59: In Exercises 5766, find the domain of the function h(t) = 4
 1.3.60: In Exercises 5766, find the domain of the function s(y) = 3y y + 5
 1.3.61: In Exercises 5766, find the domain of the function f(x) = 3 x 4
 1.3.62: In Exercises 5766, find the domain of the function f(x) = 4 x2 + 3x
 1.3.63: In Exercises 5766, find the domain of the function . g(x) = 1 x 3 x...
 1.3.64: In Exercises 5766, find the domain of the function h(x) = 10 x2 2x
 1.3.65: In Exercises 5766, find the domain of the function g(y) = y + 2 y 10
 1.3.66: In Exercises 5766, find the domain of the function . f(x) = x + 6
 1.3.67: In Exercises 67 70, use a graphing utility to graph the function. F...
 1.3.68: In Exercises 67 70, use a graphing utility to graph the function. F...
 1.3.69: In Exercises 67 70, use a graphing utility to graph the function. F...
 1.3.70: In Exercises 67 70, use a graphing utility to graph the function. F...
 1.3.71: Write the area A of a circle as a function of its circumference C.
 1.3.72: Write the area A of an equilateral triangle as a function of the le...
 1.3.73: An open box of maximum volume is to be made from a square piece of ...
 1.3.74: A right triangle is formed in the first quadrant by the x and yax...
 1.3.75: A rectangle is bounded by the xaxis and the semicircle y = 36 x2, ...
 1.3.76: A rectangular package to be sent by the U.S. Postal Service can hav...
 1.3.77: A company produces a product for which the variable cost is $68.75 ...
 1.3.78: The table shows the revenue y (in thousands of dollars) of a landsc...
 1.3.79: The total numbers n (in millions) of miles for all public roadways ...
 1.3.80: The force F (in tons) of water against the face of a dam is estimat...
 1.3.81: The height y (in feet) of a baseball thrown by a child is y = 0.1x2...
 1.3.82: The graph shows the sales (in millions of dollars) of Green Mountai...
 1.3.83: In Exercises 83 86, find the difference quotient and simplify your ...
 1.3.84: In Exercises 83 86, find the difference quotient and simplify your ...
 1.3.85: In Exercises 83 86, find the difference quotient and simplify your ...
 1.3.86: In Exercises 83 86, find the difference quotient and simplify your ...
 1.3.87: In Exercises 87 and 88, determine whether the statement is true or ...
 1.3.88: In Exercises 87 and 88, determine whether the statement is true or ...
 1.3.89: In Exercises 89 and 90, write a square root function for the graph ...
 1.3.90: In Exercises 89 and 90, write a square root function for the graph ...
 1.3.91: Given f(x) = x2, is f the independent variable? Why or why not?
 1.3.92: The graph represents the height h of a projectile after t seconds. ...
 1.3.93: In Exercises 93 96, perform the operation and simplify 2 4 x + 2
 1.3.94: In Exercises 93 96, perform the operation and simplify 2 + x 20 + 2...
 1.3.95: In Exercises 93 96, perform the operation and simplify 2x3 + 4x2 4x...
 1.3.96: In Exercises 93 96, perform the operation and simplify x + 7 2(x 9)...
Solutions for Chapter 1.3: Functions and Their Graphs
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 1.3: Functions and Their Graphs
Get Full SolutionsSince 96 problems in chapter 1.3: Functions and Their Graphs have been answered, more than 67582 students have viewed full stepbystep solutions from this chapter. Chapter 1.3: Functions and Their Graphs includes 96 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

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.

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.

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.

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.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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.