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The exercises in the image ask to find the range of the given functions for problems 17 to 20. Let's solve each function step by step:

### 17. \( f(x) = 10 - x^2 \)

This is a downward-opening parabola. The range depends on the vertex and the direction of the parabola.

- The function is of the form \( f(x) = a(x-h)^2 + k \), where \( a = -1 \), \( h = 0 \), and \( k = 10 \).
- The vertex is at \( (0, 10) \), and since \( a = -1 \), the parabola opens downward.

The maximum value of \( f(x) \) is at the vertex, where \( f(0) = 10 \). As \( x^2 \) grows larger, \( f(x) \) decreases without bound.

**Range of \( f(x) \): \( (-\infty, 10] \)**

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### 18. \( g(x) = 5 + \sqrt{4 - x} \)

For this function, the square root limits the domain since the expression under the square root, \( 4 - x \), must be non-negative.

- The condition for the square root is \( 4 - x \geq 0 \), or \( x \leq 4 \).
- The minimum value of \( g(x) \) occurs when \( x = 4 \), giving \( g(4) = 5 + 0 = 5 \).
- The maximum value occurs when \( x = -\infty \), though it approaches a limiting value.

**Range of \( g(x) \): \( [5, \infty) \)**

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Would you like to explore more details about these ranges or move on to the remaining exercises? Here are some related questions:

1. What is the vertex form of a quadratic equation?
2. How do we determine the domain of a square root function?
3. What transformations affect the range of a quadratic function?
4. How does the range of a function relate to its graph?
5. How can we restrict the domain of a function to change its range?

**Tip:** Always check the domain of a function first, especially if it involves square roots or fractions.

Find the range of the function for f(x) = 10 - x^2 and g(x) = 5 + sqrt(4 - x).

Find the range of the functions f(x) = 10 - x^2 and g(x) = 5 + sqrt(4 - x). This problem explores quadratic functions and square root functions with clear solutions and a step-by-step process.

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