 1.2.1.2.1: A relation that assigns to each element from a set of inputs, or __...
 1.2.1.2.2: For an equation that represents as a function of the _______ variab...
 1.2.1.2.3: The function is an example of a _______ function.
 1.2.1.2.4: If the domain of the function is not given, then the set of values ...
 1.2.1.2.5: In calculus, one of the basic definitions is that of a _______ , gi...
 1.2.1.2.6: In Exercises 58, decide whether the relation represents y as a func...
 1.2.1.2.7: In Exercises 58, decide whether the relation represents y as a func...
 1.2.1.2.8: In Exercises 58, decide whether the relation represents y as a func...
 1.2.1.2.9: In Exercises 9 and 10, which sets of ordered pairs represent functi...
 1.2.1.2.10: In Exercises 9 and 10, which sets of ordered pairs represent functi...
 1.2.1.2.11: Circulation of Newspapers In Exercises 11 and 12, use the graph, wh...
 1.2.1.2.12: Circulation of Newspapers In Exercises 11 and 12, use the graph, wh...
 1.2.1.2.13: In Exercises 1324, determine whether the equation represents y as a...
 1.2.1.2.14: In Exercises 1324, determine whether the equation represents y as a...
 1.2.1.2.15: In Exercises 1324, determine whether the equation represents y as a...
 1.2.1.2.16: In Exercises 1324, determine whether the equation represents y as a...
 1.2.1.2.17: In Exercises 1324, determine whether the equation represents y as a...
 1.2.1.2.18: In Exercises 1324, determine whether the equation represents y as a...
 1.2.1.2.19: In Exercises 1324, determine whether the equation represents y as a...
 1.2.1.2.20: In Exercises 1324, determine whether the equation represents y as a...
 1.2.1.2.21: In Exercises 1324, determine whether the equation represents y as a...
 1.2.1.2.22: In Exercises 1324, determine whether the equation represents y as a...
 1.2.1.2.23: In Exercises 1324, determine whether the equation represents y as a...
 1.2.1.2.24: In Exercises 1324, determine whether the equation represents y as a...
 1.2.1.2.25: In Exercises 25 and 26, fill in the blanks using the specified func...
 1.2.1.2.26: In Exercises 25 and 26, fill in the blanks using the specified func...
 1.2.1.2.27: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.28: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.29: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.30: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.31: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.32: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.33: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.34: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.35: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.36: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.37: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.38: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.39: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.40: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.41: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.42: In Exercises 27 42, evaluate the function at each specified value o...
 1.2.1.2.43: In Exercises 43 46, complete the table.
 1.2.1.2.44: In Exercises 43 46, complete the table.
 1.2.1.2.45: In Exercises 43 46, complete the table.
 1.2.1.2.46: In Exercises 43 46, complete the table.
 1.2.1.2.47: In Exercises 4750, find all real values of such thatf(x) = 0.
 1.2.1.2.48: In Exercises 4750, find all real values of such thatf(x) = 0.
 1.2.1.2.49: In Exercises 4750, find all real values of such thatf(x) = 0.
 1.2.1.2.50: In Exercises 4750, find all real values of such thatf(x) = 0.
 1.2.1.2.51: In Exercises 51 and 52, find the value(s) of for whichf x g x .
 1.2.1.2.52: In Exercises 51 and 52, find the value(s) of for whichf x g x .
 1.2.1.2.53: In Exercises 5362, find the domain of the function.
 1.2.1.2.54: In Exercises 5362, find the domain of the function.
 1.2.1.2.55: In Exercises 5362, find the domain of the function.
 1.2.1.2.56: In Exercises 5362, find the domain of the function.
 1.2.1.2.57: In Exercises 5362, find the domain of the function.
 1.2.1.2.58: In Exercises 5362, find the domain of the function.
 1.2.1.2.59: In Exercises 5362, find the domain of the function.
 1.2.1.2.60: In Exercises 5362, find the domain of the function.
 1.2.1.2.61: In Exercises 5362, find the domain of the function.
 1.2.1.2.62: In Exercises 5362, find the domain of the function.
 1.2.1.2.63: In Exercises 6366, use a graphing utility to graph the function. Fi...
 1.2.1.2.64: In Exercises 6366, use a graphing utility to graph the function. Fi...
 1.2.1.2.65: In Exercises 6366, use a graphing utility to graph the function. Fi...
 1.2.1.2.66: In Exercises 6366, use a graphing utility to graph the function. Fi...
 1.2.1.2.67: In Exercises 6770, assume that the domain of is the set Determine t...
 1.2.1.2.68: In Exercises 6770, assume that the domain of is the set Determine t...
 1.2.1.2.69: In Exercises 6770, assume that the domain of is the set Determine t...
 1.2.1.2.70: In Exercises 6770, assume that the domain of is the set Determine t...
 1.2.1.2.71: Geometry Write the area of a circle as a function of its circumfere...
 1.2.1.2.72: Geometry Write the area of an equilateral triangle as a function of...
 1.2.1.2.73: Exploration The cost per unit to produce a radio model is $60. The ...
 1.2.1.2.74: Exploration An open box of maximum volume is to be made from a squa...
 1.2.1.2.75: Geometry A right triangle is formed in the first quadrant by the an...
 1.2.1.2.76: Geometry A rectangle is bounded by the axis and the semicircle (se...
 1.2.1.2.77: Postal Regulations A rectangular package to be sent by the U.S. Pos...
 1.2.1.2.78: Cost, Revenue, and Profit A company produces a toy for which the va...
 1.2.1.2.79: Revenue In Exercises 79 82, use the table, which shows the monthly ...
 1.2.1.2.80: Revenue In Exercises 79 82, use the table, which shows the monthly ...
 1.2.1.2.81: Revenue In Exercises 79 82, use the table, which shows the monthly ...
 1.2.1.2.82: Revenue In Exercises 79 82, use the table, which shows the monthly ...
 1.2.1.2.83: Motor Vehicles The numbers (in billions) of miles traveled by vans,...
 1.2.1.2.84: Transportation For groups of 80 or more people, a charter bus compa...
 1.2.1.2.85: Physics The force (in tons) of water against the face of a dam is e...
 1.2.1.2.86: Data Analysis The graph shows the retail sales (in billions of doll...
 1.2.1.2.87: In Exercises 8792, find the difference quotient and simplify your a...
 1.2.1.2.88: In Exercises 8792, find the difference quotient and simplify your a...
 1.2.1.2.89: In Exercises 8792, find the difference quotient and simplify your a...
 1.2.1.2.90: In Exercises 8792, find the difference quotient and simplify your a...
 1.2.1.2.91: In Exercises 8792, find the difference quotient and simplify your a...
 1.2.1.2.92: In Exercises 8792, find the difference quotient and simplify your a...
 1.2.1.2.93: True or False? In Exercises 93 and 94, determine whether the statem...
 1.2.1.2.94: True or False? In Exercises 93 and 94, determine whether the statem...
 1.2.1.2.95: Library of Parent Functions In Exercises 9598, write a piecewisede...
 1.2.1.2.96: Library of Parent Functions In Exercises 9598, write a piecewisede...
 1.2.1.2.97: Library of Parent Functions In Exercises 9598, write a piecewisede...
 1.2.1.2.98: Library of Parent Functions In Exercises 9598, write a piecewisede...
 1.2.1.2.99: Writing In your own words, explain the meanings of domain and range.
 1.2.1.2.100: Think About It Describe an advantage of function notation.
 1.2.1.2.101: In Exercises 101104, perform the operation and simplify.
 1.2.1.2.102: In Exercises 101104, perform the operation and simplify.
 1.2.1.2.103: In Exercises 101104, perform the operation and simplify.
 1.2.1.2.104: In Exercises 101104, perform the operation and simplify.
Solutions for Chapter 1.2: Functions
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 1.2: Functions
Get Full SolutionsChapter 1.2: Functions includes 104 full stepbystep solutions. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Since 104 problems in chapter 1.2: Functions have been answered, more than 45941 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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).

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

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.