Recall that the absolute value |x| for a real number x is defined as the function...

Recall that the absolute value |x| for a real number x is defined as the function

|x| = x if x ≥ 0 or −x if x < 0.
Prove that, for any real numbers x and y, we have
(a) | − x| = |x|.
(b) |xy| = |x||y|.
(c) |x^-1| = 1/|x|. Here we assume that x is not equal to 0.
(d) −|x| ≤ x ≤ |x|.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Find the absolute maximum value and absolute minimum value, if any, of the function g(x) =...

Find the absolute maximum value and absolute minimum value, if any, of the function g(x) = x √(4 − x2) on the interval [−2, 2].

Find the absolute maximum value and the absolute minimum value of the function f(x,y) = (1+x^2)(1−y^2)...

Find the absolute maximum value and the absolute minimum value of the function f(x,y) = (1+x^2)(1−y^2) on the disk D = {(x,y) | x2+y2⩽1}?

1) Suppose that a function f(x) is defined for all real values of x, except x...

1) Suppose that a function f(x) is defined for all real values of x, except x = xo. Can anything be said about lim x → x 0 f ( x ) ? Give reasons for your answer. 2) If x4 ≤ f(x) ≤ x2 for x in [-1,1] and x2 ≤ f(x) ≤ x4 for x < -1 and x > 1, at what points c do you automatically know lim x → c f ( x ) ?...

What is the absolute max / min value for the function f(x) = x sqrt 1...

What is the absolute max / min value for the function f(x) = x sqrt 1 - x on the interval [ 1 , 1 ]

The function f(x) = x^2 (x^2 − 2x − 36) is defined on all real numbers....

The function f(x) = x^2 (x^2 − 2x − 36) is defined on all real numbers. On what subset of the real numbers is f(x) concave down?

The function f(x,y,z)=4x−9y+7z has an absolute maximum value and absolute minimum value subject to the constraint...

The function f(x,y,z)=4x−9y+7z has an absolute maximum value and absolute minimum value subject to the constraint x^2+y^2+z^2=146. Use Lagrange multipliers to find these values.

Find the absolute maximum and the absolute minimum of the function f(x,y) = 6 - x²...

Find the absolute maximum and the absolute minimum of the function f(x,y) = 6 - x² - y² over the region R = {(x,y) | -2 <= x <= 2, -1 <= y <= 1 }. Also mention the points at which the maximum and minimum will occur.

Use Newton's method to find the absolute maximum value of the function f(x) = 3x cos(x),...

Use Newton's method to find the absolute maximum value of the function f(x) = 3x cos(x), 0 ≤ x ≤ π; correct to six decimal places.

1) Show the absolute value function f(x) = |x| is continuous at every point. 2) Suppose...

1) Show the absolute value function f(x) = |x| is continuous at every point. 2) Suppose A and B are sets then define the cartesian product A * B Please answer both the questions.

Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be...

Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be the usual “less than or equal to” total order on the set A=(−2,0)∪(0,2) Consider the subset B={−1/n | n∈N, n≥1}⊆A. Determine an upper bound for B in A. Then formally prove that B has no least upper bound in A by arguing that every element of A fails the criteria in the definition of least upper bound. Note: least upper bound is an upper...

Subjects