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The task is to find the range of the functions in Exercises 17–20. Let's solve them one by one.

### 17. \( f(x) = 10 - x^2 \)
This is a downward-opening parabola.

- The vertex occurs at \( x = 0 \), where \( f(0) = 10 \).
- As \( x^2 \) increases, \( f(x) \) decreases without bound as \( x \to \pm \infty \).

Thus, the range is:
\[
(-\infty, 10]
\]

### 18. \( g(x) = 5 + \sqrt{4 - x} \)
Here, the function involves a square root, which requires that \( 4 - x \geq 0 \), or \( x \leq 4 \).

- At \( x = 4 \), \( g(x) = 5 + \sqrt{4 - 4} = 5 \).
- As \( x \to -\infty \), \( g(x) \) increases.

Thus, the range is:
\[
[5, \infty)
\]

### 19. \( f(x) = \frac{x^2}{1 - x^2} \)
This is a rational function with vertical asymptotes where the denominator is zero, i.e., \( 1 - x^2 = 0 \), or \( x = \pm 1 \).

- As \( x \to \pm 1 \), \( f(x) \to \infty \).
- For \( x^2 \to 0 \), \( f(x) \to 0 \).
- As \( x^2 \to \infty \), \( f(x) \to -1 \).

Thus, the range is:
\[
(-\infty, -1) \cup (0, \infty)
\]

### 20. \( g(x) = \frac{3 + x^2}{4 - x^2} \)
This is another rational function with vertical asymptotes where \( 4 - x^2 = 0 \), or \( x = \pm 2 \).

- As \( x \to \pm 2 \), \( g(x) \to \infty \).
- For \( x = 0 \), \( g(0) = \frac{3}{4} \).
- As \( x^2 \to 4 \), \( g(x) \to \pm \infty \).

Thus, the range is:
\[
(-\infty, \infty)
\]

Would you like further details on any of these solutions?

### 5 Related Questions:
1. How can you determine the domain of a function with a square root?
2. Why do rational functions often have asymptotes?
3. What is the importance of the vertex in a quadratic function's range?
4. How can we simplify complex rational expressions to find their behavior at infinity?
5. What happens to a function as its denominator approaches zero?

**Tip:** Always check the domain restrictions first when working with functions that involve square roots or rational expressions.

In Exercises 17-20, find the range of the function.

Explore how to find the range of functions including quadratic, square root, and rational functions in this detailed solution. Perfect for high school students in grades 10-12 learning about advanced algebra.

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