(A) How are the graphs of y = x + 5 and y = x − 4 related to the graph of y = x ? Confirm your answer by graphing all three functions simultaneously in the same coordinate system . (B) How are the graphs of y = x + 5 and y = x − 4 related to the graph of y = x ? Confirm your answer by graphing all three functions simultaneously in the same coordinate system.
(A) How are the graphs of y = x + 5 and y = x − 4 related to the graph of y = x ? Confirm your answer by graphing all three functions simultaneously in the same coordinate system . (B) How are the graphs of y = x + 5 and y = x − 4 related to the graph of y = x ? Confirm your answer by graphing all three functions simultaneously in the same coordinate system.
Solution Summary: The author explains how to determine the relation between graphs of the functions y=sqrtx+5, Y = . They also explain the answer by graphing all three functions
(A) How are the graphs of
y
=
x
+
5
and
y
=
x
−
4
related to the graph of
y
=
x
? Confirm your answer by graphing all three functions simultaneously in the same coordinate system.
(B) How are the graphs of
y
=
x
+
5
and
y
=
x
−
4
related to the graph of
y
=
x
? Confirm your answer by graphing all three functions simultaneously in the same coordinate system.
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
Let A be a vector space with basis 1, a, b. Which (if any) of the following rules
turn A into an algebra? (You may assume that 1 is a unit.)
(i) a² = a, b² = ab = ba = 0.
(ii) a²=b, b² = ab = ba = 0.
(iii) a²=b, b² = b, ab = ba = 0.
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= 1. Show
(a) Let G = Z/nZ be a cyclic group, so G = {1, 9, 92,...,g" } with g":
that the group algebra KG has a presentation KG = K(X)/(X” — 1).
(b) Let A = K[X] be the algebra of polynomials in X. Let V be the A-module
with vector space K2 and where the action of X is given by the matrix
Compute End(V) in the cases
(i) x = p,
(ii) xμl.
(67) ·
(c) If M and N are submodules of a module L, prove that there is an isomorphism
M/MON (M+N)/N.
(The Second Isomorphism Theorem for modules.)
You may assume that MON is a submodule of M, M + N is a submodule of L
and the First Isomorphism Theorem for modules.
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Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
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