 3.1.1: What is the difference between a relation and a function?
 3.1.2: What is the difference between the input and the output of a function?
 3.1.3: Why does the vertical line test tell us whether the graph of a rela...
 3.1.4: How can you determine if a relation is a onetoone function?
 3.1.5: Why does the horizontal line test tell us whether the graph of a fu...
 3.1.6: For the following exercises, determine whether the relation represe...
 3.1.7: For the following exercises, determine whether the relation represe...
 3.1.8: For the following exercises, determine whether the relation represe...
 3.1.9: For the following exercises, determine whether the relation represe...
 3.1.10: For the following exercises, determine whether the relation represe...
 3.1.11: For the following exercises, determine whether the relation represe...
 3.1.12: For the following exercises, determine whether the relation represe...
 3.1.13: For the following exercises, determine whether the relation represe...
 3.1.14: For the following exercises, determine whether the relation represe...
 3.1.15: For the following exercises, determine whether the relation represe...
 3.1.16: For the following exercises, determine whether the relation represe...
 3.1.17: For the following exercises, determine whether the relation represe...
 3.1.18: For the following exercises, determine whether the relation represe...
 3.1.19: For the following exercises, determine whether the relation represe...
 3.1.20: For the following exercises, determine whether the relation represe...
 3.1.21: For the following exercises, determine whether the relation represe...
 3.1.22: For the following exercises, determine whether the relation represe...
 3.1.23: For the following exercises, determine whether the relation represe...
 3.1.24: For the following exercises, determine whether the relation represe...
 3.1.25: For the following exercises, determine whether the relation represe...
 3.1.26: For the following exercises, determine whether the relation represe...
 3.1.27: For the following exercises, evaluate the function f at the indicat...
 3.1.28: For the following exercises, evaluate the function f at the indicat...
 3.1.29: For the following exercises, evaluate the function f at the indicat...
 3.1.30: For the following exercises, evaluate the function f at the indicat...
 3.1.31: For the following exercises, evaluate the function f at the indicat...
 3.1.32: Given the function g(x) = 5 x2 , simplify g(x + h) g(x) __ h , h 0
 3.1.33: Given the function g(x) = x2 + 2x, simplify _ g(x) g(a) x a , x a
 3.1.34: Given the function k(t) = 2t 1: a. Evaluate k(2). b. Solve k(t) = 7.
 3.1.35: Given the function f(x) = 8 3x: a. Evaluate f(2). b. Solve f(x) = 1
 3.1.36: Given the function p(c) = c2 + c: a. Evaluate p(3). b. Solve p(c) = 2.
 3.1.37: Given the function f(x) = x 2 3x a. Evaluate f(5). b. Solve f(x) = 4
 3.1.38: Given the function f(x) = x + 2 : a. Evaluate f(7). b. Solve f(x) = 4
 3.1.39: Consider the relationship 3r + 2t = 18. a. Write the relationship a...
 3.1.40: For the following exercises, use the vertical line test to determin...
 3.1.41: For the following exercises, use the vertical line test to determin...
 3.1.42: For the following exercises, use the vertical line test to determin...
 3.1.43: For the following exercises, use the vertical line test to determin...
 3.1.44: For the following exercises, use the vertical line test to determin...
 3.1.45: For the following exercises, use the vertical line test to determin...
 3.1.46: For the following exercises, use the vertical line test to determin...
 3.1.47: For the following exercises, use the vertical line test to determin...
 3.1.48: For the following exercises, use the vertical line test to determin...
 3.1.49: For the following exercises, use the vertical line test to determin...
 3.1.50: For the following exercises, use the vertical line test to determin...
 3.1.51: For the following exercises, use the vertical line test to determin...
 3.1.52: Given the following graph a. Evaluate f(1). b. Solve for f(x) = 3.
 3.1.53: Given the following graph a. Evaluate f(0). b. Solve for f(x) = 3.
 3.1.54: Given the following graph a. Evaluate f(4). b. Solve for f(x) = 1.
 3.1.55: For the following exercises, determine if the given graph is a one...
 3.1.56: For the following exercises, determine if the given graph is a one...
 3.1.57: For the following exercises, determine if the given graph is a one...
 3.1.58: For the following exercises, determine if the given graph is a one...
 3.1.59: For the following exercises, determine if the given graph is a one...
 3.1.60: For the following exercises, determine whether the relation represe...
 3.1.61: For the following exercises, determine whether the relation represe...
 3.1.62: For the following exercises, determine whether the relation represe...
 3.1.63: For the following exercises, determine if the relation represented ...
 3.1.64: For the following exercises, determine if the relation represented ...
 3.1.65: For the following exercises, determine if the relation represented ...
 3.1.66: For the following exercises, use the function f represented in Tabl...
 3.1.67: For the following exercises, use the function f represented in Tabl...
 3.1.68: For the following exercises, evaluate the function f at the values ...
 3.1.69: For the following exercises, evaluate the function f at the values ...
 3.1.70: For the following exercises, evaluate the function f at the values ...
 3.1.71: For the following exercises, evaluate the function f at the values ...
 3.1.72: For the following exercises, evaluate the function f at the values ...
 3.1.73: For the following exercises, evaluate the function f at the values ...
 3.1.74: For the following exercises, evaluate the expressions, given functi...
 3.1.75: For the following exercises, evaluate the expressions, given functi...
 3.1.76: For the following exercises, graph y = x2 on the given viewing wind...
 3.1.77: For the following exercises, graph y = x2 on the given viewing wind...
 3.1.78: For the following exercises, graph y = x2 on the given viewing wind...
 3.1.79: For the following exercises, graph y = x3 on the given viewing wind...
 3.1.80: For the following exercises, graph y = x3 on the given viewing wind...
 3.1.81: For the following exercises, graph y = x3 on the given viewing wind...
 3.1.82: For the following exercises, graph y = x on the given viewing windo...
 3.1.83: For the following exercises, graph y = x on the given viewing windo...
 3.1.84: For the following exercises, graph y = x on the given viewing windo...
 3.1.85: For the following exercises, graph y = 3 x on the given viewing win...
 3.1.86: For the following exercises, graph y = 3 x on the given viewing win...
 3.1.87: For the following exercises, graph y = 3 x on the given viewing win...
 3.1.88: The amount of garbage, G, produced by a city with population p is g...
 3.1.89: The number of cubic yards of dirt, D, needed to cover a garden with...
 3.1.90: Let f(t) be the number of ducks in a lake t years after 1990. Expla...
 3.1.91: Let h(t) be the height above ground, in feet, of a rocket t seconds...
 3.1.92: Show that the function f(x) = 3(x 5)2 + 7 is not onetoone.
Solutions for Chapter 3.1: Functions and Function Notation
Full solutions for College Algebra  1st Edition
ISBN: 9781938168383
Solutions for Chapter 3.1: Functions and Function Notation
Get Full SolutionsSince 92 problems in chapter 3.1: Functions and Function Notation have been answered, more than 34740 students have viewed full stepbystep solutions from this chapter. Chapter 3.1: Functions and Function Notation includes 92 full stepbystep solutions. This textbook survival guide was created for the textbook: College Algebra, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. College Algebra was written by and is associated to the ISBN: 9781938168383.

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

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.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

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.

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

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

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

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