Quadratic & Linear Function problem

Hello!

I have a more complex problem but for simplifications purposes I will explain it in simpler terms:

I have a function which I know its behaviour:

$\displaystyle y= x^2$ (IF x<a) OR $\displaystyle y=x$ (IF x>a).

Note that this is a single, continuum function. I have 2 problems in here:

1. I dont know how to write this down in an analytical formula. I want a general formula that gives me the expression of y and includes the conditions that if x<a or x>a;

2. What I want to plot is the crossover from the quadratic regime to the linear regime. That is, i want an analytical formula that gives me the expression of the crossover.


Could I have some help please? All tips are welcomed!
Many thanks

Quote:

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Originally Posted by Sosi

Hello!

I have a more complex problem but for simplifications purposes I will explain it in simpler terms:

I have a function which I know its behaviour:

$\displaystyle y= x^2$ (IF x<a) OR $\displaystyle y=x$ (IF x>a).

Note that this is a single, continuum function. I have 2 problems in here:

1. I dont know how to write this down in an analytical formula. I want a general formula that gives me the expression of y and includes the conditions that if x<a or x>a;

2. What I want to plot is the crossover from the quadratic regime to the linear regime. That is, i want an analytical formula that gives me the expression of the crossover.


Could I have some help please? All tips are welcomed!
Many thanks

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What you have is

$\displaystyle f(x)=\left\{\begin{array}{cc}x^2,&\mbox{ if }
x\leq a\\x, & \mbox{ if } x>a\end{array}\right.$.

To determine the value of $\displaystyle a$. find any points of intersection...

do you think there is any way of writing down the formula of y without using \left\{\ ? I mean, a more regular way with only the basic operational symbols ( *, /, + or - )

Quote:

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Originally Posted by Sosi

do you think there is any way of writing down the formula of y without using \left\{\ ? I mean, a more regular way with only the basic operational symbols ( *, /, + or - )

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No. this is a piecewise defined function by its very nature.