Composite function of a piecewise function

In summary, the function has two parts, one that is between 0 and 1/2 and one that is between 1/2 and 3/4. If x is between those two boundaries, then f(x) is between those two values.
  • #1
libragirl79
31
0

Homework Statement



Given that I have a doubling function :
f(x)=2x (for 0≤x<0.5) and 2x-1 (for 0.5≤ x<1)

Homework Equations


What is f(f(x))?


The Attempt at a Solution


f(f(x))=4x for the first one and 4x-3 for the second part but not sure what to do about the domain constraints...

Thanks!
 
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  • #2
libragirl79 said:

Homework Statement



Given that I have a doubling function :
f(x)=2x (for 0≤x<0.5) and 2x-1 (for 0.5≤ x<1)

Homework Equations


What is f(f(x))?


The Attempt at a Solution


f(f(x))=4x for the first one and 4x-3 for the second part but not sure what to do about the domain constraints...

Thanks!
It's not that simple.

Yes, f(f(x)) = 4x over part of the domain of the composite function and 4x-3 over some other portion, but those are not the only two pieces of f(f(x)).


For what values of x is 0 ≤ 2x < 0.5 ?

For what values of x is 0.5 ≤ 2x < 1 ?

etc.
 
  • #3
right, that's exactly my issue, since i don't have the fcns for 1/4 ≤ x < 1/2
and 1/2 ≤ x < 3/4 ... is there a certain method for doing this?
 
  • #4
libragirl79 said:
right, that's exactly my issue, since i don't have the fcns for 1/4 ≤ x < 1/2
and 1/2 ≤ x < 3/4 ... is there a certain method for doing this?
You understand that "[itex]1/4\le x< 1/2[/itex]" is part of the interval [itex]0\le x< 1/2[/itex] don't you?

If [itex]0\le x< 1/4[/itex], f(x)= 2x which is less than 1/2 so ff(x)= f(2x)= 2(2x)= 4x.
If [itex]1/4\le x< 1/2[/itex] then x is still between 0 and 1/2 so f(x)= 2x but f(x) is now between 1/2 and 1 so ff(x)= f(2x)= 2(2x)- 1= 4x- 1.

Do similarly for [itex]1/2\le x< 3/4[/itex]. Now x is between 1/2 and 1 so f(x)= 2x- 1 which is between 0 and 1/2.

If x is between 3/4 and 1, f(x)= 2x- 1 is between 1/2 and 1
 
  • #5
I understand the breakdown of the x domains, but how do you know where f(x) falls?
 

1. What is the definition of a composite function?

A composite function is a mathematical function that is formed by combining two or more functions. The output of one function is used as the input for another function.

2. How is a composite function of a piecewise function evaluated?

To evaluate a composite function of a piecewise function, you must first determine which piece of the function the input value falls into, and then apply the corresponding function to that input value.

3. What is the role of the domain in a composite function of a piecewise function?

The domain of a composite function of a piecewise function is the set of all possible input values for which the function is defined. It is important to consider the domain when evaluating a composite function to ensure that the input values are valid.

4. How is a composite function of a piecewise function graphed?

To graph a composite function of a piecewise function, you can graph each individual piece of the function separately and then combine the graphs. It is important to pay attention to any discontinuities or changes in the function at the points where the pieces meet.

5. What are some real-world applications of composite functions of piecewise functions?

Composite functions of piecewise functions are commonly used in engineering, physics, and economics to model complex systems and relationships. For example, they can be used to calculate the optimal production levels for a company or to model the motion of a projectile with changing forces acting on it.

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