Absolute value and piecewise functions

In summary, absolute value is a mathematical concept that represents the distance of a number from zero on a number line, always resulting in a positive value. It is represented by two vertical bars in equations. To graph an absolute value function, you must plot the vertex and two other points, and then connect them with a V-shaped curve. Piecewise functions are used when a function's behavior changes at certain points or intervals, and they are evaluated by determining which rule applies to the given input. Absolute value and piecewise functions can be combined in one equation, where the absolute value function may be defined by different rules and may also be part of a larger piecewise function with multiple rules.
  • #1
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Homework Statement



Find if continuous and differentiable. I am having problems with the differentiable part.


Homework Equations



f(x) = x² + 3, |x| ≤ 1
f(x) = |x| + 3, |x| > 1

The Attempt at a Solution



(1)^2 + 3 = 4
|1| + 3 = 4
∴ It is continuous

Now, rewriting it, you have...
x² + 3, x ≥ -1 or x ≤ 1
-x + 3, x < -1 or x > 1
x + 3, x < -1 or x > 1

Finding the limits from left and right of 1...
From the left = 2x, = 2
From the right: can be both -1 and 1...

Where did I go wrong?
 
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  • #2
939 said:

Homework Statement



Find if continuous and differentiable. I am having problems with the differentiable part.


Homework Equations



f(x) = x² + 3, |x| ≤ 1
f(x) = |x| + 3, |x| > 1

The Attempt at a Solution



(1)^2 + 3 = 4
|1| + 3 = 4
∴ It is continuous

Now, rewriting it, you have...
x² + 3, x ≥ -1 or x ≤ 1
-x + 3, x < -1 or x > 1
x + 3, x < -1 or x > 1

Finding the limits from left and right of 1...
From the left = 2x, = 2
From the right: can be both -1 and 1...

Where did I go wrong?

No, you've got
x² + 3, x ≥ -1 or x ≤ 1
-x + 3, x < -1
x + 3, x > 1

Worry about the two point x=1 and x=(-1) separately.
 
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  • #3
Dick said:
No, you've got
x² + 3, x ≥ -1 or x ≤ 1
-x + 3, x < -1
x + 3, x > 1

Worry about the two point x=1 and x=(-1) separately.

Thanks a lot!

Final question can |(2/3)(x)| be turned into (2/3)(-x) and (2/3)(x)?
 
  • #4
939 said:
Thanks a lot!

Final question can |(2/3)(x)| be turned into (2/3)(-x) and (2/3)(x)?

Sure. |2x/3| is -2x/3 if x<0 and 2x/3 if x>=0.
 
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Likes 1 person

1. What is absolute value and how is it represented in mathematical equations?

Absolute value is a mathematical concept that represents the distance of a number from zero on a number line. It is always positive, regardless of the sign of the number. In equations, absolute value is represented by two vertical bars surrounding the number or expression.

2. How do you graph absolute value functions?

To graph an absolute value function, plot the vertex (the point where the absolute value expression equals zero) and two additional points on either side of the vertex. Then, draw a V-shaped curve connecting the points. If the number inside the absolute value is negative, the graph will be reflected over the x-axis.

3. What is a piecewise function and when is it used?

A piecewise function is a function that is defined by different rules or equations for different parts of its domain. It is typically used when the function's behavior changes at certain points or intervals. For example, a piecewise function may have one rule for x < 0 and a different rule for x ≥ 0.

4. How do you evaluate a piecewise function?

To evaluate a piecewise function, you need to determine which rule or equation applies to the given input. For example, if the input is 2 and the function has one rule for x < 0 and another rule for x ≥ 0, the rule for x ≥ 0 would apply. Then, plug the input into the corresponding rule to find the output.

5. Can absolute value and piecewise functions be combined in one equation?

Yes, absolute value and piecewise functions can be combined in one equation. For example, an absolute value function may be defined by two different rules for x < 0 and x ≥ 0, making it a piecewise function. Additionally, the absolute value function may also be part of a larger piecewise function with multiple rules for different parts of the domain.

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