 9.1: a. Sketch the graphs of the following functions on the same grid: f...
 9.9.1.1: In each part of the problem the graph of (x) to the left has been t...
 9.9.2.1: Given (t) 5 3t 2 1 4t 2 5 and g(t) 5 6t 1 1, find: a. (t) 1 g(t) b....
 9.9.3.1: . Identify which of the following are polynomial functions and, for...
 9.9.4.1: If the cost for a producer to produce n items is C(n) 1000 100n, th...
 9.9.5.1: From the accompanying table, find: a. f(g(1)) c. f(g(0)) e. f( f(2)...
 9.2: Using the three accompanying graphs of polynomial functions in the ...
 9.9.1.2: Match each of the following functions with its graph. Identify the ...
 9.9.2.2: Given and , find and simplify: a. (m) 1 g(m) c. ( g )(m) e. b. ( 2 ...
 9.9.3.2: Identify the degree of any of the following functions that are poly...
 9.9.4.2: The startup costs for a small pizza company are $100,000 (the fixe...
 9.9.5.2: From the accompanying table, find: a. f(g(1)) c. f(g(0)) e. f( f(2)...
 9.3: Which of the following statements are true about the graph of the p...
 9.9.1.3: Let (x) x3 . a. Write the equation for the new function g(x) that r...
 9.9.2.3: Let (x) 5 3x5 1 x and g(x) 5 x2 2 1. a. Construct the following fun...
 9.9.3.3: Evaluate the following expressions for x 5 2 and x 5 22. a. x23 b. ...
 9.9.4.3: Use a calculator to evaluate each expression at x 10, x 100, and x ...
 9.9.5.3: Using the accompanying graphs, find: a. g(f(2)) b. f(g(21)) c. g(f(...
 9.4: a. Generate two different polynomials, M(z) and N(z), that have hor...
 9.9.1.4: Explain in words the effect of the following transformations on the...
 9.9.2.4: If h(x) (x) g(x) x2 3x 4, what are possible equations for (x) and g...
 9.9.3.4: Evaluate the following polynomials for x 5 2 and x 5 22, and specif...
 9.9.4.4: Use a calculator to evaluate each function at x 10, x 100, and x 10...
 9.9.5.4: Using the accompanying graphs, find: a. g(f(22)) b. f(g(1)) c. g(f(...
 9.5: On the same grid, handdraw rough sketches of the three functions f...
 9.9.1.5: Decide if each graph (although not necessarily a function) is symme...
 9.9.2.5: You own a theater company and you have an upcoming event. a. You de...
 9.9.3.5: Match each of the following functions with its graph at the top of ...
 9.9.4.5: Determine the coordinates of the point(s) where the graph of the fu...
 9.9.5.5: Given F(x) 5 2x 1 1 and , find: a. F(G(1)) d. F(F(0)) b. G(F(22)) e...
 9.6: a. Given the following graph of the function f(x), sketch: i. g(x) ...
 9.9.1.6: Complete the partial graph shown on the next page in three differen...
 9.9.2.6: Many colleges around the country are finding they need to buy more ...
 9.9.3.6: Match each of the following functions with its graph. a. f(x) 5 x2 ...
 9.9.4.6: If , find f(x) 2f(x 3); then find a common denominator and combine ...
 9.9.5.6: Given f(x) 5 3x 2 2 and g(x) 5 (x 1 1)2 , find: a. f(g(1)) d. f(f(2...
 9.7: (Graphing program required.) a. If and , describe the transformatio...
 9.9.1.7: (Graphing program optional.) A function is said to be even if (x) (...
 9.9.2.7: A worker gets $20/hour for a normal work week of 40 hours and time...
 9.9.3.7: For each of the graphs of polynomial functions, at the top of the n...
 9.9.4.7: For each rational function graphed below, estimate the equation for...
 9.9.5.7: The winds are calm, allowing a forest fire to spread in a circular ...
 9.8: Global warming melts glaciers and polar ice, so scientists predict ...
 9.9.1.8: Use the function (x) to create a new function g(x) where the graph ...
 9.9.2.8: Using the accompanying table, evaluate the following expressions in...
 9.9.3.8: Describe how g(x) and h(x) relate to (x). (x) 5 x5 2 3x2 1 4 g(x) 5...
 9.9.4.8: (Graphing program optional.) For each of the functions in Exercises...
 9.9.5.8: The exchange rate a bank gave for Canadian dollars on June 27, 2010...
 9.9: Given the graph of f(x) and g(x) below: Find the following values a...
 9.9.1.9: For each function, construct a new function whose graph is the grap...
 9.9.2.9: Use the table in Exercise 8 to create a new table for the functions...
 9.9.3.9: Divide the following functions into groups having the same global s...
 9.9.4.9: (Graphing program optional.) For each of the functions in Exercises...
 9.9.5.9: A stone is dropped into a pond, causing a circular ripple that is e...
 9.10: If f(x) (1/2)x and g(x) (1/2) x (x 4), a. Find expressions for i. f...
 9.9.1.10: If write the equation for g(x) that represents each of the followin...
 9.9.2.10: (Graphing program optional.) One method of graphing functions is ca...
 9.9.3.10: Estimate the maximum number of turning points for each of the polyn...
 9.9.4.10: (Graphing program optional.) For each of the functions in Exercises...
 9.9.5.10: The wind chill temperature is the apparent temperature caused by th...
 9.11: Retirement fund counselors often recommend a mixed portfolio of inv...
 9.9.1.11: Write an equation for each function, g, h, and j, based on f(x) x. ...
 9.9.2.11: Using the accompanying graph of f(x) and g(x), find estimates for t...
 9.9.3.11: (Graphing program optional.) Describe the behavior of each polynomi...
 9.9.4.11: (Graphing program required.) a. What is the domain of the rational ...
 9.9.5.11: Salt is applied to roads to decrease the temperature at which icing...
 9.12: A typical retirement scheme for state employees is based on three t...
 9.9.1.12: Write an equation for each function, g, h, and j, based on f (x) x2...
 9.9.2.12: From the graph and your results in Exercise 11, find the equations ...
 9.9.3.12: (Graphing program required.) Find the vertical intercept and estima...
 9.9.4.12: (Graphing program required.) Let a. We can think of g(x) as being c...
 9.9.5.12: Using the given functions f, g, and h where f(x) x 1 g(x) 5 ex h(x)...
 9.13: For the graphs below, use function notation, to express g(x) and h(...
 9.9.1.13: The equation for Graph A is . Assuming all of the graphs are the sa...
 9.9.2.13: The Richland Banquet Hall charges $500 to rent its facility and $40...
 9.9.3.13: (Graphing program required.) Find the vertical intercept and estima...
 9.9.4.13: (Graphing program required.) Construct a rational function f(x) tha...
 9.9.5.13: Using the given functions J, K, and L, where J(x) x3 K(x) log(x) L(...
 9.9.1.14: If h(x) f(x 5), a. What changes are made to the input and output of...
 9.14: Motorola has taken steps to diversify its business by introducing m...
 9.9.2.14: Given the following graphs of f(x) and g(x): a. Draw the graph of (...
 9.9.3.14: a. (Graphing program required.) Use a function graphing program to ...
 9.9.4.14: The function f(x) 1/x was transformed into the function g(x) plotte...
 9.9.5.14: In Exercises 14 and 15, rewrite j(x) as the composition of three fu...
 9.9.1.15: Apply the transformations specified in parts (a)(e) to f(x) ln x. f...
 9.15: Match each of the following graphs with its function, find the equa...
 9.9.2.15: The following are graphs of h(x) and j(x). Without using technology...
 9.9.3.15: (Graphing program required.) Use a function graphing program (and i...
 9.9.4.15: Without using technology, match each function with its graph. a. f(...
 9.9.5.15: In Exercises 14 and 15, rewrite j(x) as the composition of three fu...
 9.9.1.16: (Graphing program optional.) a. Starting with the function f(x) ex ...
 9.16: Given the functions f(x) x2 , g(x) x 1, and h(x) 3x, evaluate each ...
 9.9.2.16: The accompanying graph gives the production P( y), consumption C(y)...
 9.9.3.16: (Graphing program required.) Identify the xintercepts of the follo...
 9.9.4.16: (Graphing program required for part (c).) The rational function can...
 9.9.5.16: In Exercises 1622, show that the two functions are inverses of each...
 9.9.1.17: Given function f(x) 10 5x , find the function g(x) if: a. The graph...
 9.17: Most current refrigerators (such as the 2010 Energy Star) cost abou...
 9.9.2.17: Almost all organizations find it necessary to maintain inventories ...
 9.9.3.17: a. If the degree of a polynomial is odd, then at least one of its z...
 9.9.4.17: If , a. Describe the transformations of f(x) used to create the new...
 9.9.5.17: In Exercises 1622, show that the two functions are inverses of each...
 9.9.1.18: The solidline graphs below represent exponential functions in the ...
 9.18: (Graphing program required.) Given the rational function , determin...
 9.9.2.18: In Chapter 7 we learned about even and odd power functions and in S...
 9.9.3.18: In each part, construct a polynomial function with the indicated ch...
 9.9.4.18: Let . Construct a new function j(s) that is the end result of the t...
 9.9.5.18: In Exercises 1622, show that the two functions are inverses of each...
 9.9.1.19: a. Given describe the transformations that created Find b. Use your...
 9.19: Does the function f(x) (x 2)3 1 have an inverse? If not, explain wh...
 9.9.2.19: Use the definition of even and odd functions in Exercise 18 to veri...
 9.9.3.19: Polynomial expressions of the form an3 bn2 cn d can be used to expr...
 9.9.4.19: (Graphing program required.) You live in a house that borders a riv...
 9.9.5.19: In Exercises 1622, show that the two functions are inverses of each...
 9.9.1.20: The following two graphs show the hours of daylight during the year...
 9.20: Which of these functions has an inverse? If there is one, what is it?
 9.9.2.20: The accompanying graph gives the annual sales S(t) and profit (or l...
 9.9.3.20: (Graphing program required.) A manufacturer sells childrens wooden ...
 9.9.4.20: A student drives nonstop from Missoula, Montana to Spokane, Washin...
 9.9.5.20: In Exercises 1622, show that the two functions are inverses of each...
 9.9.1.21: (Graphing program required.) If an object is put in an environment ...
 9.21: When lightning strikes, you seem to see it right away, but the asso...
 9.9.2.21: In Section 1.2, p. 17, Exercise 1, there is a graph about AIDS diag...
 9.9.3.21: Factor the following to determine the horizontal intercepts. Then d...
 9.9.4.21: The graduate math department at an East Coast university subsidizes...
 9.9.5.21: In Exercises 1622, show that the two functions are inverses of each...
 9.9.1.22: (Graphing program required.) Newtons Law of Cooling (see Exercise 2...
 9.22: The Texas Cancer Center website, www.texascancercenter.com, notes t...
 9.9.2.22: When considering a career path in a particular job sector, one migh...
 9.9.3.22: Factor the following to determine the horizontal intercepts. Then d...
 9.9.4.22: Building with energysaving walls becomes increasingly important as...
 9.9.5.22: In Exercises 1622, show that the two functions are inverses of each...
 9.9.1.23: In the accompanying figure, match the graphs labeled A, B, and C wi...
 9.23: For 1725 give an example of a function or functions with the specif...
 9.9.3.23: Construct a polynomial function with the following xintercepts and...
 9.9.5.23: In Exercises 23 and 24, create a table of values for the inverse of...
 9.9.1.24: The accompanying graph shows the height of sunflowers over time. As...
 9.24: For 1725 give an example of a function or functions with the specif...
 9.9.3.24: Construct three thirddegree polynomials, each of which has horizon...
 9.9.5.24: In Exercises 23 and 24, create a table of values for the inverse of...
 9.25: For 1725 give an example of a function or functions with the specif...
 9.9.3.25: Construct a polynomial function for each of the following sets of p...
 9.9.5.25: Cryptology (the creation and deciphering of codes) is based on 11 ...
 9.26: In 2008 the population of C2(t) was approximately half the populati...
 9.9.3.26: Are the only functions with xintercepts at 0, 2, and 1 of the form...
 9.9.5.26: On June 27, 2010, the conversion rate from U.S. dollars to euros wa...
 9.27: C1(t) C2(t) is a constant.
 9.9.3.27: Match each polynomial function with its graph. f(x) (x 3)2 (x 2)2 g...
 9.9.5.27: Given the accompanying graph of f(x), answer the following. a. Does...
 9.28: A polynomial function of degree 4 will always have three turning po...
 9.9.3.28: An ice bucket is designed as a cubic block of foam with a centered ...
 9.9.5.28: Determine which of the accompanying graphs show functions that are ...
 9.29: A polynomial function of degree 4 will cross the horizontal axis ex...
 9.9.3.29: Professional fundraisers typically establish fundraising categori...
 9.9.5.29: In Exercises 2931, for each function Q find Q1 , if it exists. For ...
 9.30: A polynomial function of odd degree must cross the horizontal axis ...
 9.9.3.30: (Graphing program required.) Buildings lose heat in three different...
 9.9.5.30: In Exercises 2931, for each function Q find Q1 , if it exists. For ...
 9.31: The leading term determines the global shape of the graph of a poly...
 9.9.5.31: In Exercises 2931, for each function Q find Q1 , if it exists. For ...
 9.32: The three polynomial functions in the accompanying figure are all o...
 9.9.5.32: Use the graph of f(x) to estimate the value of each expression. a. ...
 9.33: The functions f(x) ln x and g(x) ex are inverses of each other.
 9.9.5.33: The following tables represent a function f that converts cups to q...
 9.34: f 21(x) 5 1 f(x) for any function f
 9.9.5.34: Let f(x) mx b. a. Does f(x) always have an inverse? Explain. b. If ...
 9.35: If and g(x) 2x2 1, then (f + g)(x) 5 1 (2x 2 1 1)
 9.9.5.35: If you do an Internet search on formulas for ideal body weight (IBW...
 9.36: If f(x) f(x), the graph of f is symmetric across the xaxis.
 9.9.5.36: The formula for the volume of a cone is V 5 r2 h. Assume you are ho...
 9.37: If f(x) f(x), the graph of f is symmetric about the origin. 3
 9.9.5.37: In Chapter 6 we learned that a logarithm can be constructed using a...
 9.9.5.38: The percentage of a building exterior that is glass is very importa...
 9.38: A function that passes the vertical line test has an inverse
 9.39: Every function is 11.
 9.40: The graph below shows the linear path of a ray of light, P, and the...
Solutions for Chapter 9: NEW FUNCTIONS FROM OLD
Full solutions for Explorations in College Algebra  5th Edition
ISBN: 9780470466445
Solutions for Chapter 9: NEW FUNCTIONS FROM OLD
Get Full SolutionsThis textbook survival guide was created for the textbook: Explorations in College Algebra, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Since 176 problems in chapter 9: NEW FUNCTIONS FROM OLD have been answered, more than 8262 students have viewed full stepbystep solutions from this chapter. Explorations in College Algebra was written by and is associated to the ISBN: 9780470466445. Chapter 9: NEW FUNCTIONS FROM OLD includes 176 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

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).

Outer product uv T
= column times row = rank one matrix.

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

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.

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

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

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