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.