I need help creating an equation for a piecewise defined function with two different rules, that is continuous but not differentiable at the point where the rules change, x=3. Please help to justify the answer.
To create a piecewise defined function that is continuous but not differentiable at the point where the rules change (x = 3), you can follow these steps:
Step 1: Determine the rules for each part of the function.
Since the function needs to have two different rules, let's say we have rule A for x < 3 and rule B for x > 3.
Step 2: Create the equation for each part of the function.
For rule A, let's choose a simple linear equation that is continuous but not differentiable. One possible equation is:
f(x) = 2x - 6
For rule B, we need to choose another equation that is also continuous but not differentiable at x = 3. Let's say:
f(x) = x^2 - 9
Step 3: Combine the equations with piecewise notation.
Now, to combine the two equations into a single piecewise defined function, we can use the following notation:
f(x) = {2x - 6, if x < 3
{x^2 - 9, if x > 3
The function has two different rules defined based on the value of x being less than or greater than 3. At x = 3, the function changes from rule A to rule B, creating a jump or discontinuity in the function, which makes it continuous but not differentiable at that point.
Remember, the justification for the answer lies in the fact that the function has a change in rules at x = 3, resulting in a jump or discontinuity, which prevents the function from being differentiable at x = 3.