Explore BrainMass
Share

Real analysis

This content was STOLEN from BrainMass.com - View the original, and get the already-completed solution here!

A. let f(x):= 1/(x^2), x does not equal zero, x elements of Reals
determine the direct image f(E) where E= {x elements Reals: 1<=x<=2}
determine the inverse image f^-1(G) where G= { x element of Reals: 1<=x<=4}

b. show that if a<b then a< 1/2(a+b)<b

c. for a,b,c elements of reals with a<b, find an explicit bijection of A:={x:a<x<b} onto B:={y: 0<y<1}, show how you derived.

d. if a,b elements of Reals show absval(a+b) = abs(a) + abs(b) if and only if ab >= 0

e. Find all x elements of reals that satisfy the following inequalities
i. abs(x-1) > abs(x+1) ii. abs(x) + abs(x+1) < 2

© BrainMass Inc. brainmass.com October 25, 2018, 12:26 am ad1c9bdddf
https://brainmass.com/math/real-analysis/228020

Solution Preview

(a) since f is decreasing f(E) = { y : 1/4 =< y =< 1 };

f^{-1}(G) = {x in R : f(x) in G } = {x in R : 1/2 = < x =< 1}

(b) if ab >= 0, then either (a >= 0 and b >= 0) or (a =< 0 and b =< 0);

case 1: both a, b >= 0; then a + b >= 0, so |a + b| = a + b = |a| + |b|;

case 2 : both a, b =< 0; then a + b =< 0 and |a| = -a and |b| = -b, so |a + b| = -(a + b) = -a + (-b) = |a| + |b|

Conversely, assume |a + b| = |a| + |b|; then on ...

Solution Summary

This provides examples of working with analysis, including bijections, absolute value, and images.

$2.19
See Also This Related BrainMass Solution

Regression model for real estate data

Refer to the data included in the Excel file, which report information on homes sold in the Somewhere, USA, during a recent year. Use the selling price of the home as the dependent variable and determine the regression equation with number of bedrooms, size of the house, whether there is a pool, whether there is an attached garage, distance from the center of the
city, and the number of bathrooms as independent variables.

i) Construct a 95% confidence interval estimate of the population slope between selling price and each of the following variables: number of bedrooms, size in sq. ft., distance to CBD, and number of bathrooms.
j) Compute and interpret the coefficients of partial determination.
k) Predict the selling price of a 2,500 square feet house that has 5 bedrooms, 3 bathrooms,
a 3-car attached garage, no pool, and is at 18 miles from the city center.
l) Rerun the analysis until only significant regression coefficients remain in the analysis. Identify these variables.

View Full Posting Details