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The exercise in the image asks to find the domain and range of the following functions:

1. \( f(x) = 1 + x^2 \)
2. \( f(x) = 1 - \sqrt{x} \)
3. \( F(x) = \sqrt{5x + 10} \)
4. \( g(x) = \sqrt{x^2 - 3x} \)
5. \( f(t) = \frac{4}{3 - t} \)
6. \( G(t) = \frac{2}{t^2 - 16} \)

### Let's solve each one for domain and range:

1. **\( f(x) = 1 + x^2 \)**
   - **Domain**: \( x \in (-\infty, \infty) \), since the quadratic function is defined for all real numbers.
   - **Range**: \( y \geq 1 \), as \( x^2 \) is always non-negative, and the minimum value is \(1\) when \(x = 0\).

2. **\( f(x) = 1 - \sqrt{x} \)**
   - **Domain**: \( x \geq 0 \), because the square root function is defined for non-negative values.
   - **Range**: \( y \leq 1 \), since \( \sqrt{x} \geq 0 \), the maximum value occurs when \( x = 0 \), and the function decreases as \( x \) increases.

3. **\( F(x) = \sqrt{5x + 10} \)**
   - **Domain**: \( 5x + 10 \geq 0 \) ⟶ \( x \geq -2 \).
   - **Range**: \( y \geq 0 \), since the square root function always returns non-negative values.

4. **\( g(x) = \sqrt{x^2 - 3x} \)**
   - **Domain**: \( x^2 - 3x \geq 0 \) ⟶ \( x(x - 3) \geq 0 \), which gives \( x \leq 0 \) or \( x \geq 3 \) by solving the inequality.
   - **Range**: \( y \geq 0 \), since it's a square root function.

5. **\( f(t) = \frac{4}{3 - t} \)**
   - **Domain**: \( 3 - t \neq 0 \) ⟶ \( t \neq 3 \).
   - **Range**: All real numbers except \( y = 0 \), because the function never equals zero.

6. **\( G(t) = \frac{2}{t^2 - 16} \)**
   - **Domain**: \( t^2 - 16 \neq 0 \) ⟶ \( t \neq \pm 4 \).
   - **Range**: All real numbers except \( y = 0 \), as the denominator never makes the function zero.

Would you like further details or explanations?

### Related Questions:
1. How do you find the domain of a square root function?
2. What is the range of a quadratic function?
3. How does the domain of a rational function differ from that of a square root function?
4. Can the range of a rational function include zero? Why or why not?
5. How do you solve inequalities to find the domain of a function?

**Tip**: Always check for values that make the denominator zero or cause square roots to produce negative values when determining the domain.

In Exercises 1-6, find the domain and range of each function.
1. f(x) = 1 + x^2
2. f(x) = 1 - sqrt(x)
3. F(x) = sqrt(5x + 10)
4. g(x) = sqrt(x^2 - 3x)
5. f(t) = 4 / (3 - t)
6. G(t) = 2 / (t^2 - 16)

This exercise asks students to find the domain and range of six different functions, including quadratic, square root, and rational functions. Topics covered include determining when functions are undefined, solving inequalities, and analyzing square root and rational equations. Suitable for Grades 10-12.

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