Find the domain and range of the functions:
1. f(x)=3x
Domain:
Range:
2. f(x)=x²-6
Domain:
Range:
3. f(x)=x²-1
Domain:
Range:
4. f(x)=4/x+2
Domain:
Range:
Show solutions.

Let's determine the domain and range for each of the functions step by step.

### Function 1: \( f(x) = 3x \)
- **Domain**: 
  - Since \( f(x) = 3x \) is a linear function, there are no restrictions on \( x \). Any real number can be input into the function.
  - **Domain**: \( \mathbb{R} \) (All real numbers)

- **Range**:
  - The range of a linear function \( f(x) = ax + b \) (where \( a \neq 0 \)) is all real numbers because as \( x \) varies over all real numbers, \( f(x) \) takes all real values.
  - **Range**: \( \mathbb{R} \) (All real numbers)

### Function 2: \( f(x) = x^2 - 6 \)
- **Domain**:
  - The function \( f(x) = x^2 - 6 \) is a polynomial, so it is defined for all real numbers.
  - **Domain**: \( \mathbb{R} \) (All real numbers)

- **Range**:
  - The expression \( x^2 \) is always non-negative, meaning \( x^2 \geq 0 \). Thus, \( x^2 - 6 \geq -6 \).
  - The smallest value of \( f(x) \) occurs when \( x = 0 \), giving \( f(x) = -6 \).
  - **Range**: \( [-6, \infty) \)

### Function 3: \( f(x) = x^2 - 1 \)
- **Domain**:
  - Similarly, \( f(x) = x^2 - 1 \) is a polynomial, so it is defined for all real numbers.
  - **Domain**: \( \mathbb{R} \) (All real numbers)

- **Range**:
  - Since \( x^2 \geq 0 \), we have \( x^2 - 1 \geq -1 \).
  - The smallest value of \( f(x) \) occurs when \( x = 0 \), giving \( f(x) = -1 \).
  - **Range**: \( [-1, \infty) \)

### Function 4: \( f(x) = \frac{4}{x+2} \)
- **Domain**:
  - The function is undefined when the denominator is zero, i.e., \( x + 2 = 0 \) which gives \( x = -2 \).
  - **Domain**: \( \mathbb{R} \setminus \{-2\} \) (All real numbers except \( x = -2 \))

- **Range**:
  - \( f(x) = \frac{4}{x+2} \) can take all real values except 0, because there is no value of \( x \) that will make \( f(x) = 0 \).
  - **Range**: \( \mathbb{R} \setminus \{0\} \) (All real numbers except 0)

### Summary
1. \( f(x) = 3x \)  
   - **Domain**: \( \mathbb{R} \)  
   - **Range**: \( \mathbb{R} \)

2. \( f(x) = x^2 - 6 \)  
   - **Domain**: \( \mathbb{R} \)  
   - **Range**: \( [-6, \infty) \)

3. \( f(x) = x^2 - 1 \)  
   - **Domain**: \( \mathbb{R} \)  
   - **Range**: \( [-1, \infty) \)

4. \( f(x) = \frac{4}{x+2} \)  
   - **Domain**: \( \mathbb{R} \setminus \{-2\} \)  
   - **Range**: \( \mathbb{R} \setminus \{0\} \)

Would you like further details on any of these, or have any other questions?

Here are 5 related questions to deepen your understanding:
1. What is the domain of the function \( f(x) = \sqrt{x+3} \)?
2. Find the range of the function \( f(x) = 2x^2 + 5x - 3 \).
3. Determine the domain of \( f(x) = \frac{1}{x^2 - 4} \).
4. What is the range of \( f(x) = \sin(x) \)?
5. Find the domain and range of the function \( f(x) = |x-1| \).

**Tip:** When determining the range of a quadratic function, consider its vertex, as it provides the minimum or maximum value depending on the parabola's direction.

Discover how to determine the domain and range of functions such as f(x)=3x, f(x)=x²-6, f(x)=x²-1, and f(x)=4/(x+2). Learn the rules and steps to find the domain and range for each function with detailed explanations and examples. Suitable for high school students and anyone studying functions in mathematics.

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