Difference between revisions of "Absolute Value Functions"

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==Introduction==
 
==Introduction==
[[Image:Absolute value.svg|thumb|The [[Graphs|graph]] of the absolute value function for real numbers]]
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[[Image:Absolute value.svg|thumb|The graph of the absolute value function for real numbers]]
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[[File:Graph abs(x^2-4).png|thumb|The graph of the function y = |x^2 - 4|]]
 
In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. Namely, |x| = x if x is positive, and |x| = −x if x is negative (in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its "distance from zero". The absolute value function f(x) = |a| can be expressed as a piecewise function, where f(x) = a when a ≥ 0, and f(x) = -a when a < 0. We can use this to help us visualize, graph, or solve other absolute value functions.
 
In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. Namely, |x| = x if x is positive, and |x| = −x if x is negative (in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its "distance from zero". The absolute value function f(x) = |a| can be expressed as a piecewise function, where f(x) = a when a ≥ 0, and f(x) = -a when a < 0. We can use this to help us visualize, graph, or solve other absolute value functions.
  
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* <math> g(x) = |5x + 5| </math>. 5x + 5 ≥ 0 when x ≥ -1, and 5x + 5 < 0 when x < -1. So, g(x) = 5x + 5 when x ≥ -1, and g(x) = -5x - 5 when x < -1.  
 
* <math> g(x) = |5x + 5| </math>. 5x + 5 ≥ 0 when x ≥ -1, and 5x + 5 < 0 when x < -1. So, g(x) = 5x + 5 when x ≥ -1, and g(x) = -5x - 5 when x < -1.  
 
* <math> h(x) = |x^2 - 4| </math>. <math> x^2 - 4 \ge 0 </math> when -2 ≥ x or x ≥ 2. So, <math> h(x) = x^2 - 4 </math> when -2 ≥ x or x ≥ 2, and <math> h(x) = -x^2 + 4 </math> when -2 < x < 2.
 
* <math> h(x) = |x^2 - 4| </math>. <math> x^2 - 4 \ge 0 </math> when -2 ≥ x or x ≥ 2. So, <math> h(x) = x^2 - 4 </math> when -2 ≥ x or x ≥ 2, and <math> h(x) = -x^2 + 4 </math> when -2 < x < 2.
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Abs(x^2-5).svg
  
 
==Resources==
 
==Resources==

Revision as of 12:10, 15 September 2021

Introduction

The graph of the absolute value function for real numbers
Graph abs(x^2-4).png

In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. Namely, |x| = x if x is positive, and |x| = −x if x is negative (in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its "distance from zero". The absolute value function f(x) = |a| can be expressed as a piecewise function, where f(x) = a when a ≥ 0, and f(x) = -a when a < 0. We can use this to help us visualize, graph, or solve other absolute value functions.

Here are some examples of absolute value functions:

  • . 5x + 5 ≥ 0 when x ≥ -1, and 5x + 5 < 0 when x < -1. So, g(x) = 5x + 5 when x ≥ -1, and g(x) = -5x - 5 when x < -1.
  • . when -2 ≥ x or x ≥ 2. So, when -2 ≥ x or x ≥ 2, and when -2 < x < 2.

Abs(x^2-5).svg

Resources