Help with piecewise functions: find "a" so y = 2x-1 for x <=4, a+5x, x>4, is continuous

[GSM]

I have some piecewise functions that I am hoping someone can help me with because I cannot find a solution and I am confused with the pronumerals? The question is "Determine the values for the pronumerals that make the following piecewise functions continuous."

[math]y = \begin{cases} 2x - 1, \hspace{2mm} x \le 4 \\ a+5x, \hspace{2mm} x>4 \end{cases}[/math]
[math]y = \begin{cases} -x, \hspace{5mm} x \le a \\ 2+x, \hspace{2mm} a < x \le b \\ 2x - 5, \hspace{1mm} x > b \end{cases}[/math]
Thanks in advance.

 

[stapel]

Determine the values for the pronumerals that make the following piecewise functions continuous.

What does it mean for a function to be "continuous"?

[imath]\qquad 1)\; y = \begin{cases} 2x - 1, \hspace{2mm} x \le 4 \\ a+5x, \hspace{2mm} x>4 \end{cases}[/imath]

What is the value of 2x-1 at x = 4? What is the value of a+5x at x = 4? Where does this lead?

[imath]\qquad 2)\; y = \begin{cases} -x, \hspace{5mm} x \le a \\ 2+x, \hspace{2mm} a < x \le b \\ 2x - 5, \hspace{1mm} x > b \end{cases}[/imath]

Are you sure that the above is a correct statement of the exercise?

When you reply, please include a clear listing of your thoughts and efforts so far. Thank you!

Eliz.

 

[Dr.Peterson]

I have some piecewise functions that I am hoping someone can help me with because I cannot find a solution and I am confused with the pronumerals? The question is "Determine the values for the pronumerals that make the following piecewise functions continuous."

[math]y = \begin{cases} 2x - 1, \hspace{2mm} x \le 4 \\ a+5x, \hspace{2mm} x>4 \end{cases}[/math]
[math]y = \begin{cases} -x, \hspace{5mm} x \le a \\ 2+x, \hspace{2mm} a < x \le b \\ 2x - 5, \hspace{1mm} x > b \end{cases}[/math]
Thanks in advance.

The second is different from most that I see like this; I'd start by graphing each of the three subfunctions without regard to their boundaries. (That is, graph 3 entire lines.)

 

[GSM]

Thanks for your replies. I have figured out the first piecewise function but I am still having trouble with the second one. I know the 2 values that need to be connected are a and b but how do I find their value? All the examples / videos I have studied have numbers on the right hand side of the functions and not pronumerals?

Yes the second function "is a correct statement of the exercise?"

Thank you for your help.

 

[Dr.Peterson]

Thanks for your replies. I have figured out the first piecewise function but I am still having trouble with the second one. I know the 2 values that need to be connected are a and b but how do I find their value? All the examples / videos I have studied have numbers on the right hand side of the functions and not pronumerals?

Yes the second function "is a correct statement of the exercise?"

Thank you for your help.

Did you try doing what I suggested? As I said, this one is different from most examples you'll see, so you have to think for yourself. And the way to start is to just graph y = -x, y = 2 + x, and y = 2x - 5.

 

[stapel]

Thanks for your replies. I have figured out the first piecewise function but I am still having trouble with the second one. I know the 2 values that need to be connected are a and b but how do I find their value? All the examples / videos I have studied have numbers on the right hand side of the functions and not pronumerals?

[math]y = \begin{cases} -x, \hspace{5mm} x \le a \\ 2+x, \hspace{2mm} a < x \le b \\ 2x - 5, \hspace{1mm} x > b \end{cases}[/math]
Yes the second function "is a correct statement of the exercise"

For the second function to be continuous, the pieces have to meet (that is, they must have matching values) at the cut-offs, being [imath]a[/imath] and [imath]b[/imath]. So what happens when you plug in the "values" for the matching parts?

Eliz.

 

[pka]

I have some piecewise functions that I am hoping someone can help me with because I cannot find a solution and I am confused with the pronumerals? The question is "Determine the values for the pronumerals that make the following piecewise functions continuous."
[imath]y = \begin{cases} 2x - 1, \hspace{2mm} x \le 4 \\ a+5x, \hspace{2mm} x>4 \end{cases}[/imath]
[imath]y = \begin{cases} -x, \hspace{5mm} x \le a \\ 2+x, \hspace{2mm} a < x \le b \\ 2x - 5, \hspace{1mm} x > b \end{cases}[/imath]

As I see this, there are two distinct problems.
In the first you need to find [imath]\bf a[/imath] that is, [imath]\mathop {\lim }\limits_{x \to {\bf 4 - }} \left( {2x - 1} \right) = \mathop {\lim }\limits_{x \to {4^ + }} \left( {a + 5x} \right)[/imath]

Now for the second one you need to find [imath]\bf{a~\&~b}[/imath], that is:
[imath]\mathop {\lim }\limits_{x \to a - } \left( { - x} \right) = \mathop {\lim }\limits_{x \to a + } \left( {2 + x} \right)\quad \& \quad \mathop {\lim }\limits_{x \to b - } \left( {2 + x} \right) = \mathop {\lim }\limits_{x \to b + } \left( {2x - 5} \right)[/imath]