 2.1.1: Any set of ordered pairs is called a/an ______________. The set of ...
 2.1.2: A set of ordered pairs in which each member of the set of first com...
 2.1.3: The notation f(x) describes the value of ______________ at ________...
 2.1.4: If h(r) = r2 + 4r  7, we can find h(2) by replacing each occurren...
 2.1.5: {(3, 3),(2, 2),(1, 1),(0, 0)}
 2.1.6: {(7, 7), (5, 5), (3, 3), (0, 0)}
 2.1.7: {(1, 4), (1, 5), (1, 6)}
 2.1.8: {(4, 1), (5, 1), (6, 1)}
 2.1.9: f(x) = x + 1 a. f(0) b. f(5) c. f(8) d. f(2a) e. f(a + 2)
 2.1.10: f(x) = x + 3 a. f(0) b. f(5) c. f(8) d. f(2a) e. f(a + 2)
 2.1.11: g(x) = 3x  2 a. g(0) b. g(5) c. ga 2 3 b d. g(4b) e. g(b + 4)
 2.1.12: g(x) = 4x  3 a. g(0) b. g(5) c. ga 3 4 b d. g(5b) e. g(b + 5)
 2.1.13: h(x) = 3x2 + 5 a. h(0) b. h(1) c. h(4) d. h(3) e. h(4b)
 2.1.14: h(x) = 2x2  4 a. h(0) b. h(1) c. h(5) d. h(3) e. h(5b)
 2.1.15: f(x) = 2x2 + 3x  1 a. f(0) b. f(3) c. f(4) d. f(b) e. f(5a)
 2.1.16: f(x) = 3x2 + 4x  2 a. f(0) b. f(3) c. f(5) d. f(b) e. f(5a)
 2.1.17: f(x) = (x) 3  x2  x + 7 a. f(0) b. f(2) c. f(2) d. f(1) + f(1)
 2.1.18: f(x) = (x) 3  x2  x + 10 a. f(0) b. f(2) c. f(2) d. f(1) + f(1)
 2.1.19: f(x) = 2x  3 x  4 a. f(0) b. f(3) c. f(4) d. f(5) e. f(a + h) f...
 2.1.20: f(x) = 3x  1 x  5 a. f(0) b. f(3) c. f(3) d. f(10) e. f(a + h) f...
 2.1.21: x f(x) 4 3 2 6 0 9 2 12 4 15 a. f(2) b. f(2) c. For what value o...
 2.1.22: x f(x) 5 4 3 8 0 12 3 16 5 20 a. f(3) b. f(3) c. For what value ...
 2.1.23: x h(x) 2 2 1 1 0 0 1 1 2 2 a. h(2) b. h(1) c. For what values of...
 2.1.24: x h(x) 2 2 1 1 0 0 1 1 2 2 a. h(2) b. h(1) c. For what value...
 2.1.25: Find g(1) and f(g(1)).
 2.1.26: Find g(1) and f(g(1)).
 2.1.27: Find 2f(1)  f(0)  [g(2)] 2 + f(2) , g(2) # g(1).
 2.1.28: Find 0 f(1)  f(0)0  [g(1)] 2 + g(1) , f(1) # g(2).
 2.1.29: f(x) = x3 + x  5
 2.1.30: f(x) = x2  3x + 7
 2.1.31: f(x) = b 3x + 5 if x 6 0 4x + 7 if x 0 a. f(2) b. f(0) c. f(3) d. ...
 2.1.32: f(x) = b 6x  1 if x 6 0 7x + 3 if x 0 a. f(3) b. f(0) c. f(4) d. ...
 2.1.33: a. Write a set of four ordered pairs in which each of the least cor...
 2.1.34: a. Write a set of four ordered pairs in which each of the most corr...
 2.1.35: What is a relation? Describe what is meant by its domain and its ra...
 2.1.36: Explain how to determine whether a relation is a function. What is ...
 2.1.37: Does f(x) mean f times x when referring to function f ? If not, wha...
 2.1.38: For people filing a single return, federal income tax is a function...
 2.1.39: Todays temperature is a function of the time of day.
 2.1.40: My height is a function of my age.
 2.1.41: Although I presented my function as a set of ordered pairs, I could...
 2.1.42: My function models how the chance of divorce depends on the number ...
 2.1.43: All relations are functions.
 2.1.44: No two ordered pairs of a function can have the same second compone...
 2.1.45: The domain of f = the range of f
 2.1.46: The range of f = the domain of g
 2.1.47: f(4)  f(2) = 2
 2.1.48: g(4) + f(4) = 0
 2.1.49: If f(x) = 3x + 7, find f(a + h)  f(a) h
 2.1.50: Give an example of a relation with the following characteristics: T...
 2.1.51: If f(x + y) = f(x) + f(y) and f(1) = 3, find f(2), f(3), and f(4). ...
 2.1.52: Simplify: 24 , 4[2  (5  2)] 2  6. (Section 1.2, Example 7)
 2.1.53: Simplify: 3x2y2 y3 2 . (Section 1.6, Example 9)
 2.1.54: Solve: x 3 = 3x 5 + 4. (Section 1.4, Example 4)
 2.1.55: Graph y = 2x. Select integers for x, starting with 2 and ending wi...
 2.1.56: Graph y = 2x + 4. Select integers for x, starting with 2 and endin...
 2.1.57: Use the following graph to solve this exercise. 1 1 2 3 4 5 6 5 4 3...
Solutions for Chapter 2.1: Introduction to Functions
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 2.1: Introduction to Functions
Get Full SolutionsSince 57 problems in chapter 2.1: Introduction to Functions have been answered, more than 10633 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.1: Introduction to Functions includes 57 full stepbystep solutions. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6.

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

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.

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.

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.

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

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 ().

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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

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.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).
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