Introduction to Discontinuity

Understanding the behavior of mathematical functions, especially where they are discontinuous, is crucial for advanced mathematics and engineering applications. Today, we will delve into the specific function ( f(f(x)) ) and determine the number of values of ( x ) for which it is discontinuous. This will be done by analyzing the function and breaking it down stepwise.

Function Definition

Let's start by defining the given function for ease of notation:

Step 1: Define the Inner Function

To make the problem more manageable, we will introduce an inner function ( g(x) ) such that:

[ g(x) f(f(x)) ]

For simplicity and clarity, we can define ( g(x) ) as follows:

Step 2: Define the Inner Function g(x)

[ g(x) -6 quad text{for} quad x 2 ]

[ g(x) 1 quad text{for} quad x 9 ]

[ g(x) frac{6x-5}{9-x} quad text{otherwise} ]

Determining Discontinuities

To find where ( g(x) ), and consequently ( f(f(x)) ), is discontinuous, we need to analyze the given function ( g(x) ) more closely. Discontinuities can occur due to the following reasons:

Disconnected pieces: Pieces of the function that come together at specific points. Asymptotes: Vertical or horizontal lines where the function becomes undefined or infinite. Removable discontinuities: Where the limit exists but the function is not defined or has a different value. Jump discontinuities: Where the left and right limits exist but are not equal.

Case 1: Behavior at Discrete Points

First, we observe the behavior at the discrete points where ( x ) is explicitly defined:

At ( x 2 ):

[ g(2) -6 ]

This is a defined value and we should check if it causes a discontinuity. Typically, a discrete jump here would suggest a discontinuity, but since this is the only specific value provided, we need to check further.

At ( x 9 ):

[ g(9) 1 ]

Similar to the previous point, this is a defined value and no discontinuity is implied.

Case 2: Behavior in the Integral Interval

The function ( g(x) frac{6x-5}{9-x} ) is defined for all ( x ) except where the denominator is zero:

[ 9 - x 0 Rightarrow x 9 ]

At ( x 9 ), the function becomes undefined, creating a vertical asymptote. This is a clear point of discontinuity.

Case 3: Checking for Jump Discontinuities

To identify jump discontinuities, we need to check the limits as ( x ) approaches the points of change:

[ lim_{x to 2^-} g(x) quad text{and} quad lim_{x to 2^ } g(x) ] [ lim_{x to 9^-} g(x) quad text{and} quad lim_{x to 9^ } g(x) ]

For ( x to 2 ):

[ text{Since } g(x) text{ is explicitly defined at } x 2, text{ we don't have discontinuities at } x 2. ]

For ( x to 9 ):

[ lim_{x to 9^-} frac{6x-5}{9-x} -infty ]

[ lim_{x to 9^ } frac{6x-5}{9-x} infty ]

Since the limits from the left and right of ( x 9 ) approach different infinities, there is a jump discontinuity at ( x 9 ).

Conclusion

From the analysis, we find that the function ( g(x) f(f(x)) ) is discontinuous at one point, specifically at ( x 9 ).

Number of Discontinuities: 1

Thus, the answer is that the function ( f(f(x)) ) is discontinuous at one value of ( x ).