Find all points of discontinuity for the function f(x)=(x^2−16)/(x−4)⋅(x−5) and classify them.

To find all points of discontinuity for the function \( f(x) = \frac{(x^2 - 16)}{(x - 4) \cdot (x - 5)} \), we need to look for values of \( x \) where the function is either undefined or has a discontinuous behavior. Let's go through the steps to analyze this.

### Step 1: Factor the numerator
First, factor the numerator \( x^2 - 16 \), which is a difference of squares:

\[
x^2 - 16 = (x - 4)(x + 4)
\]

So, the function becomes:

\[
f(x) = \frac{(x - 4)(x + 4)}{(x - 4)(x - 5)}
\]

### Step 2: Simplify the function
Now, we simplify the expression. We can cancel the common factor \( (x - 4) \), but note that \( x = 4 \) is a point where the original function was undefined. After simplifying, the function becomes:

\[
f(x) = \frac{x + 4}{x - 5}, \quad \text{for} \ x \neq 4
\]

### Step 3: Find points of discontinuity
The points of discontinuity occur where the function is undefined:

1. **At \( x = 4 \)**:  
   - Before simplification, the function was undefined at \( x = 4 \). Since the factor \( (x - 4) \) cancels out, this is a **removable discontinuity** (also called a hole). The function would be continuous at \( x = 4 \) if we define \( f(4) \) appropriately. 

2. **At \( x = 5 \)**:  
   - The denominator \( (x - 5) \) causes the function to be undefined at \( x = 5 \). This is a **non-removable discontinuity** (also called an infinite discontinuity) because the function blows up at this point.

### Step 4: Classification of discontinuities
- **Removable discontinuity** at \( x = 4 \).
- **Non-removable (infinite) discontinuity** at \( x = 5 \).

### Conclusion
The function \( f(x) = \frac{(x^2 - 16)}{(x - 4)(x - 5)} \) has:
- A **removable discontinuity** at \( x = 4 \).
- A **non-removable (infinite) discontinuity** at \( x = 5 \).

Would you like more details or have any questions?

### Related Questions:
1. How do you determine whether a discontinuity is removable or non-removable?
2. What is the behavior of the function \( f(x) \) as \( x \) approaches 5?
3. Can we redefine \( f(x) \) at \( x = 4 \) to make the function continuous?
4. What is the difference between a hole and an asymptote in terms of discontinuities?
5. How does the cancellation of terms affect the continuity of a rational function?

**Tip**: Always check for common factors in the numerator and denominator when finding discontinuities in rational functions. This helps identify removable discontinuities.

Find all points of discontinuity for the function f(x)=(x^2−16)/(x−4)⋅(x−5) and classify them.

Discover how to find and classify points of discontinuity for the function f(x) = (x^2−16)/(x−4)(x−5). Learn about removable and non-removable discontinuities using algebraic factorization. This problem is suitable for students in Grades 10-12.

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