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.