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In summary, the conversation is about a piecewise function with different equations for different ranges of x. The goal is to find the values of a and b in order to make the function continuous at x=2 and x=3. The first attempt at solving for a and b resulted in two equations, and another equation can be found by considering the limit at x=3.
  • #1
SmittenWCalc
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Homework Statement



This is a piece wise function of course. f(x) =

(x2-4) / (x-2) if x is less than two.

ax2 - bx + 1 if x is greater than or equal to 2, or less than three.

4x - a + b if x is greater than or equal to three.

Homework Equations


The Attempt at a Solution



Alright, I know enough to factor the top of the fist equation and get x+2. That means when x is two, f(x) is four. We can use f(x) in this case because we are making the function continuous. I've gotten as far as plugging in this value in the second equation and getting

4 = a4 - b2 + 1

but I don't know what to do from here, or how to get the values of a and b. I think I subtract one from the right and get

3 = a4 - 2b

Now I am definitely stuck.
 
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  • #2
SmittenWCalc said:

Homework Statement



This is a piece wise function of course. f(x) =

(x2-4) / (x-2) if x is less than two.

ax2 - bx + 1 if x is greater than or equal to 2, or less than three.

4x - a + b if x is greater than or equal to three.

Homework Equations


The Attempt at a Solution



Alright, I know enough to factor the top of the fist equation and get x+2. That means when x is two, f(x) is four. We can use f(x) in this case because we are making the function continuous. I've gotten as far as plugging in this value in the second equation and getting

4 = a4 - b2 + 1

but I don't know what to do from here, or how to get the values of a and b. I think I subtract one from the right and get

3 = a4 - 2b

Now I am definitely stuck.
Written in a more useful way, your equation is

4a - 2b = 3Now, what about at x = 3? You want the function to be continuous there, as well, right? What needs to happen for f to be continuous at x = 3?

That should give you another equation so that you have a system of two equations in the unknowns a and b.
 
  • #3
SmittenWCalc said:

Homework Statement



This is a piece wise function of course. f(x) =

(x2-4) / (x-2) if x is less than two.

ax2 - bx + 1 if x is greater than or equal to 2, or less than three.

4x - a + b if x is greater than or equal to three.

Homework Equations


The Attempt at a Solution



Alright, I know enough to factor the top of the fist equation and get x+2. That means when x is two, f(x) is four. We can use f(x) in this case because we are making the function continuous. I've gotten as far as plugging in this value in the second equation and getting

4 = a4 - b2 + 1

but I don't know what to do from here, or how to get the values of a and b. I think I subtract one from the right and get

3 = a4 - 2b

Now I am definitely stuck.
Hello SmittenWCalc. Welcome to PF!

So, you have the following:
[itex]\displaystyle \lim_{x\to\,2-}f(x)=\lim_{x\to\,2-}(x+2)=4\ .[/itex]

[itex]\displaystyle \lim_{x\to\,2+}f(x)=\lim_{x\to\,2+} (ax^2+bx+1)=4a+2b+1\ .[/itex]

Do something similar at x=3 .
 

What is a piecewise function?

A piecewise function is a mathematical function that is made up of multiple smaller functions, each of which is defined over a certain interval. The different parts of the function are pieced together to form a larger function.

How do you graph a piecewise function?

To graph a piecewise function, you need to graph each individual function over its defined interval and then connect the graphs at the boundaries of the intervals. This creates a piecewise graph with different sections for each part of the function.

What is the purpose of a piecewise function?

Piecewise functions are used to model real-world situations that involve different rules or conditions. They can also be used to solve optimization problems where different equations apply to different parts of the problem.

How do you find the domain of a piecewise function?

The domain of a piecewise function is the set of all possible inputs or x-values that the function can take. To find the domain, you need to consider the domain of each individual function within the piecewise function and then find the intersection of these domains.

What are some common examples of piecewise functions?

Piecewise functions can be found in many areas of science and mathematics, such as physics, economics, and engineering. Some common examples include the step function, absolute value function, and the greatest integer function.

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