How do you define absolute value function on different intervals?

In summary, the absolute value function is a mathematical function denoted by |x|, which returns the distance of a number from zero on a number line. Its graph is a V-shaped graph with a vertex at the origin, and its domain and range are all real numbers and all non-negative real numbers, respectively. It has various real-life applications in fields such as science and technology, and it always returns a positive output, except in the case of complex numbers.
  • #1
paulmdrdo1
385
0
Define f(x)= |x+3|-|x-3| without absolute value bars piecewise in the following intervals (-∞,-3);[-3,3);[3,+∞).

this is how i do the problem,

I removed the absolute value bars first

f(x)= x+3-x+3 = 6

now i don't know how to define it piecewise. can you show me how define it correctly. thanks!
 
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  • #2
Re: absolute value function.

On the interval:

i) \(\displaystyle (-\infty,-3)\)

we have:

\(\displaystyle x+3<0\,\therefore\,|x+3|=-(x+3)\)

\(\displaystyle x-3<0\,\therefore\,|x-3|=-(x-3)\)

and so \(\displaystyle f(x)=(-(x+3))-(-(x-3))=-6\)

Can you try the other two intervals?
 

FAQ: How do you define absolute value function on different intervals?

What is the absolute value function?

The absolute value function is a mathematical function that returns the distance of a number from zero on a number line. It is denoted by the symbol |x|, where x is the input value.

How is the absolute value function represented graphically?

The graph of the absolute value function is a V-shaped graph, also known as a "V-shape" or "tent" graph. The vertex of the V-shape is located at the origin (0,0) and the arms of the V-shape extend infinitely in both directions along the x-axis.

What is the domain and range of the absolute value function?

The domain of the absolute value function is all real numbers, as every input value can be evaluated for its distance from zero. The range of the function is all non-negative real numbers, as the absolute value of any number is always positive.

What are some real-life applications of the absolute value function?

The absolute value function is used in various fields of science and technology, such as physics, engineering, and economics. It is used to calculate displacement, velocity, acceleration, and other physical quantities. In economics, it is used to calculate profit and loss, as well as to model supply and demand curves.

Can the absolute value function have negative outputs?

No, the absolute value function always returns a positive output. This is because the function is defined as the distance of a number from zero, which is always positive. However, when working with complex numbers, the absolute value function can have a negative output, as the distance of a complex number from zero can be negative in the complex plane.

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