f(x) =
x2 − 4x
x2 − 16
,

x = 4.1, 4.05, 4.01, 4.001, 4.0001,
3.9, 3.95, 3.99, 3.999, 3.9999
x	f(x)
4.1

Incorrect: Your answer is incorrect.
4.05
4.01
4.001
4.0001

x	f(x)
3.9
3.95
3.99
3.999
3.9999

The function you have is:

\[
f(x) = \frac{x^2 - 4x}{x^2 - 16}
\]

To begin, simplify the function and then calculate the values for \( f(x) \) at the given \( x \) values.

### Step 1: Simplifying the function

\[
f(x) = \frac{x^2 - 4x}{x^2 - 16}
\]

Notice that \( x^2 - 16 \) can be factored as a difference of squares:

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

Thus, the function becomes:

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

For all \( x \neq 4 \), the \( (x - 4) \) terms cancel out, leaving:

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

At \( x = 4 \), the original form results in division by zero, so the function is undefined there.

### Step 2: Calculate \( f(x) \) for the given values of \( x \)

For each of the following values, we'll compute \( f(x) = \frac{x}{x + 4} \).

1. **\( x = 4.1 \)**:

\[
f(4.1) = \frac{4.1}{4.1 + 4} = \frac{4.1}{8.1} \approx 0.50617
\]

2. **\( x = 4.05 \)**:

\[
f(4.05) = \frac{4.05}{4.05 + 4} = \frac{4.05}{8.05} \approx 0.50311
\]

3. **\( x = 4.01 \)**:

\[
f(4.01) = \frac{4.01}{4.01 + 4} = \frac{4.01}{8.01} \approx 0.50062
\]

4. **\( x = 4.001 \)**:

\[
f(4.001) = \frac{4.001}{4.001 + 4} = \frac{4.001}{8.001} \approx 0.50012
\]

5. **\( x = 4.0001 \)**:

\[
f(4.0001) = \frac{4.0001}{4.0001 + 4} = \frac{4.0001}{8.0001} \approx 0.50001
\]

Now, for values of \( x \) approaching from the left (less than 4):

6. **\( x = 3.9 \)**:

\[
f(3.9) = \frac{3.9}{3.9 + 4} = \frac{3.9}{7.9} \approx 0.49367
\]

7. **\( x = 3.95 \)**:

\[
f(3.95) = \frac{3.95}{3.95 + 4} = \frac{3.95}{7.95} \approx 0.49748
\]

8. **\( x = 3.99 \)**:

\[
f(3.99) = \frac{3.99}{3.99 + 4} = \frac{3.99}{7.99} \approx 0.49937
\]

9. **\( x = 3.999 \)**:

\[
f(3.999) = \frac{3.999}{3.999 + 4} = \frac{3.999}{7.999} \approx 0.49987
\]

10. **\( x = 3.9999 \)**:

\[
f(3.9999) = \frac{3.9999}{3.9999 + 4} = \frac{3.9999}{7.9999} \approx 0.49999
\]

### Summary of Results

| \( x \)    | \( f(x) \)          |
|------------|---------------------|
| 4.1        | 0.50617             |
| 4.05       | 0.50311             |
| 4.01       | 0.50062             |
| 4.001      | 0.50012             |
| 4.0001     | 0.50001             |
| 3.9        | 0.49367             |
| 3.95       | 0.49748             |
| 3.99       | 0.49937             |
| 3.999      | 0.49987             |
| 3.9999     | 0.49999             |

Do you need further details or clarification? Here are some related questions you might want to explore:

1. What happens at \( x = 4 \)? Why is it undefined?
2. How does the behavior of the function change as \( x \) approaches 4 from the left and right?
3. Can we graph the function and identify key features such as asymptotes?
4. What is the limit of \( f(x) \) as \( x \to 4 \)?
5. Can this function be further generalized for other values of parameters?

**Tip:** When working with rational functions, always check for points of discontinuity by factoring and simplifying the expressions.

f(x) = (x^2 − 4x) / (x^2 − 16), x = 4.1, 4.05, 4.01, 4.001, 4.0001, 3.9, 3.95, 3.99, 3.999, 3.9999

Learn how to simplify the rational function f(x) = (x^2 − 4x) / (x^2 − 16) and calculate f(x) at values approaching x = 4. This solution includes factoring, limits, and evaluating values for x approaching from both sides of the discontinuity. Suitable for students in Grades 10-12.

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