On what intervals is each function continuous?
(a) 𝑓 𝑥 = 𝑥
100 − 2𝑥
37 + 75

Let's analyze the function you provided:

\[
f(x) = \frac{x}{100 - 2x} + 37 + 75
\]

This function is composed of two main parts:
1. \( \frac{x}{100 - 2x} \) (a rational function)
2. The constants \( 37 + 75 = 112 \)

### Step-by-step analysis:

#### 1. Continuity of the rational function \( \frac{x}{100 - 2x} \):
A rational function is continuous wherever its denominator is not equal to zero. Here, the denominator is \( 100 - 2x \), so we need to find where it is zero:

\[
100 - 2x = 0 \quad \Rightarrow \quad x = \frac{100}{2} = 50
\]

Thus, the function is **not continuous** at \( x = 50 \) because the denominator becomes zero, causing a division by zero.

#### 2. The constants \( 37 + 75 = 112 \):
Constants are continuous everywhere.

### Conclusion:
The overall function \( f(x) = \frac{x}{100 - 2x} + 112 \) is continuous **everywhere except at \( x = 50 \)**, where the rational part of the function becomes undefined.

### Continuity Interval:
The function is continuous on the intervals:

\[
(-\infty, 50) \cup (50, \infty)
\]

Would you like more details or have any questions? Here are some related problems you might consider:

1. What is the behavior of \( f(x) \) as \( x \to 50 \)?
2. How would you determine the limits of \( f(x) \) at the endpoints of the intervals of continuity?
3. What happens to the value of \( f(x) \) as \( x \to \infty \) or \( x \to -\infty \)?
4. How would the intervals change if the denominator were \( 100 - x \) instead of \( 100 - 2x \)?
5. Can we modify the function to make it continuous at \( x = 50 \)?

**Tip:** Rational functions are continuous everywhere their denominators are non-zero. Keep an eye on points where the denominator can become zero.

On what intervals is each function continuous? (a) f(x) = x / (100 - 2x) + 37 + 75

Explore how to determine the intervals of continuity for the function f(x) = x / (100 - 2x) + 112. This problem involves analyzing rational functions and determining points of discontinuity. Suitable for Grades 11-12 (Pre-Calculus/Calculus).

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